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# Winternitz, Pavel

Emeritus Professor ,Retired professor

Faculty of Arts and Science - Department of Mathematics and Statistics

### Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Classification of separable superintegrable systems of order four in two dimensional Euclidean space and algebras of integrals of motion in one dimension Theses and supervised dissertations / 2019-01
Sajedi, Masoumeh
Abstract
The purpose of this thesis is to continue a systematic research on classical and quantum superintegrable systems. We present the results from two articles. In the first article, we consider a general Hamiltonian in two-dimensional Euclidean space admitting a second order and a fourth order integral of motion. The second order integral imposes the separation of variable in the potentials. We present a complete classification of all quantum potentials that are solutions of nonlinear ODEs and the corresponding integrals. New potentials expressed in terms of the third and fifth Painlevé transcendents are obtained. In the second article, we develop new constructions of two-dimensional classical and quantum superintegrable Hamiltonians with separation of variables in Cartesian coordinates. Here we construct four types of algebraic Hamiltonian systems in one dimension. We study two copies of operator algebras in one dimension and combine these two to form superintegrable systems in $E_2$. We demonstrate that all quantum and classical superintegrable systems separable in Cartesian coordinates up to order 5 are in fact reducible.

Superintégrabilité quantique avec une intégrale de mouvement de cinquième ordre Theses and supervised dissertations / 2017-10
Abouamal, Ismail
Abstract
The present thesis is part of a general program whose purpose is to obtain and classify all superintegrable systems in two dimensions with integrals of motion that are polynomials of arbitrary order (degree) N in p_1 and p_2, the two components of the linear momentum. In this study, presented in a single article, we investigate the coexistence of the Hamiltonian with a fifth-order quantum integral of motion. We derive the determining equations that guarantee the existence of this integral and we simplify them by assuming that the potentials V(x,y) are separable in Cartesian coordinates, i.e. V(x,y)=V_1(x)+V_2(y), a supposition which, together with the existence of an independent fifth order integral, also guarantees the superintegrablity of the system. We show that when both V_1(x) and V_2(y) do not satisfy any linear ODE, they are always solutions of nonlinear ODEs. These equations are found to have the Painlevé property. Most of them are solved in terms of known Painlevé transcendents or elliptic functions. The others may define new higher order Painlevé trenscendents.

Invariant discretizations of partial differential equations Theses and supervised dissertations / 2015-06
Rebelo, Raphaël
Abstract
An algorithm discretizing partial differential equations (PDEs) while preserving their Lie symmetries is provided. This is made possible by the use of discrete partial derivatives transforming as their continuous counterparts under the action of local Lie groups. In applications, many PDEs are invariant under the action of Lie point symmetries of infinite dimension designated as Lie pseudo-groups. To extend the invariant discretization method to such equations, a discretization of pseudo-groups is proposed. The pseudo-group action discretization transforms the continuous point symmetries into generalized symmetries in the discrete space. Invariant schemes are then created for a number of PDEs. In all cases, numerical tests demonstrate that invariant schemes are better approximations of their continuous equivalents than standard finite differences.

Les systèmes super intégrables d'ordre trois séparables en coordonnées paraboliques Theses and supervised dissertations / 2012-04
Abstract
This thesis is a contribution to the study of classical and quantum superintegrability in a two-dimensional Euclidean space involving a third order integral of motion. It consists of an article. Because the classifications of all separable hamiltonians into Cartesian and polar coordinates are already complete, we bring to this picture the study of those systems in parabolic coordinates. First, we derive the determinating equations of a system into parabolic coordinates, after which we solve the obtained equations in order to find integrals of order three for potentials, which allow the separations of variables into the parabolic coordinates. Finally, we prove that all the third order integrals for separable potentials in parabolic coordinates in the Euclidean space of dimension two are reducible. In the conclusion of this article, we analyze the differences between the separable potentials in Cartesian and polar coordinates and the separable potentials in parabolic coordinates. Keywords: integrability, superintegrability, classical mechanics, quantum mechanics, Hamiltonian, separation of variables, commutation.

Superintégrabilité avec séparation de variables en coordonnées polaires et intégrales du mouvement d'ordre supérieur à deux Theses and supervised dissertations / 2010-10
Tremblay, Frédérick
Abstract
In this thesis, we propose new superintegrable systems separable in polar coordinates. After the introduction, in chapter 2, we present a complete classification of all separable systems in polar coordinates which admit a third order integral in addtion to the second order one responsible for the separation of variables. New potentials expressed in terms of the sixth Painlevé transcendent and of the Weierstrass elliptic function are obtained. In chapter 3 we introduce an infinite family of integrable and exactly sovable classical and quantum systems separable in polar coordinates. This family is described in term of a parameter k. The energy spectrum and the wave functions of the quantum systems are obtained. A conjecture postulating the superintegrability of these systems is formulated and is verified for the first cases k = 1,2,3,4. The order of the integrals is 2k where k ∈ ℕ. The algebraic structure of the family of quantum systems is formulated in term of a hidden algebra where the number of generators depends on the parameter k. A quasi-exactly solvable and integrable generalization of the family of potentials is proposed. Finally in chapter 4, the classical trajectories of the family of systems are calculated for all the rational cases k ∈ ℚ. Those are expressed in term of Chebyshev polynomials. We plot the curves associated with the trajectories for k=1,2,3,4,1/2, 1/3 and 3/2. The bounded curves are closed and periodic in the two dimensional phase space. Those results obtained reinforce the possible veracity of the conjecture.

Discrétisation des équations différentielles ordinaires avec préservation de leurs symétries Theses and supervised dissertations / 2003
Cyr-Gagnon, Catherine
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Systèmes intégrables et superintégrables classiques et quantiques avec champ magnétique Theses and supervised dissertations / 2003
Bérubé, Josée
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

### Selected publications Expand allCollapse all

#### First integrals of difference equations that do not have a variational formulation

Vinternits, P., Dorodnitsyn, V. A., Kaptsov, E. I. et Kozlov, R. V., First integrals of difference equations that do not have a variational formulation 454, 627--630 (2014), , Dokl. Akad. Nauk

#### Superintegrable systems with spin induced by co-algebra symmetry

Riglioni, D., Gingras, O. et Winternitz, P., Superintegrable systems with spin induced by co-algebra symmetry 47, 122002, 12 (2014), , J. Phys. A

#### Classical and quantum superintegrability with applications

Miller, Jr., Willard, Post, Sarah et Winternitz, Pavel, Classical and quantum superintegrability with applications 46, 423001, 97 (2013), , J. Phys. A

#### Continuous symmetries of Lagrangians and exact solutions of discrete equations

V. Dorodnitsyn, R. Kozlov, and P. Winternitz, Continuous symmetries of Lagrangians and exact solutions of discrete equations , (2012), , J. Math. Phys.

#### Infinite families of superintegrable systems separable in subgroup coordinates

Lévesque, Daniel, Post, Sarah et Winternitz, Pavel, Infinite families of superintegrable systems separable in subgroup coordinates 45, 465204, 17 (2012), , J. Phys. A

#### Superintegrable systems with spin et second-order integrals of motion

Désilets, Jean-Francois, Winternitz, Pavel, Yurdu\c sen, \.Ismet, Superintegrable systems with spin et second-order integrals of motion 45, 475201, 26 (2012), , J. Phys. A

#### Third-order superintegrable systems separable in parabolic coordinates

Popper, I., Post, S. et Winternitz, P., Third-order superintegrable systems separable in parabolic coordinates 53, 062105, 20 (2012), , J. Math. Phys.

#### Solvable Lie algebras with Borel nilradicals

\v Snobl, L. et Winternitz, P., Solvable Lie algebras with Borel nilradicals 45, 095202, 18 (2012), , J. Phys. A

#### Contact transformations for difference schemes

Levi, Decio, Scimiterna, Christian, Thomova, Zora et Winternitz, Pavel, Contact transformations for difference schemes 45, 022001, 9 (2012), , J. Phys. A

#### Symmetries of the continuous and discrete Krichever-Novikov equation

Levi, Decio, Winternitz, Pavel et Yamilov, Ravil I., Symmetries of the continuous and discrete Krichever-Novikov equation 7, Paper 097, 16 (2011), , SIGMA Symmetry Integrability Geom. Methods Appl.

#### Are there contact transformations for discrete equations?

Levi, Decio, Thomova, Zora et Winternitz, Pavel, Are there contact transformations for discrete equations? 44, 265201, 7 (2011), , J. Phys. A

#### A nonseparable quantum superintegrable system in 2D real Euclidean space

Post, Sarah et Winternitz, Pavel, A nonseparable quantum superintegrable system in 2D real Euclidean space 44, 162001, 8 (2011), , J. Phys. A

#### Lie point symmetries of differential-difference equations

Levi, D., Winternitz, P. et Yamilov, R. I., Lie point symmetries of differential-difference equations 43, 292002, 14 (2010), , J. Phys. A

#### Third-order superintegrable systems separating in polar coordinates

Tremblay, Frédérick et Winternitz, Pavel, Third-order superintegrable systems separating in polar coordinates 43, 175206, 17 (2010), , J. Phys. A

#### An infinite family of superintegrable deformations of the Coulomb potential

Post, Sarah et Winternitz, Pavel, An infinite family of superintegrable deformations of the Coulomb potential 43, 222001, 11 (2010), , J. Phys. A

#### Periodic orbits for an infinite family of classical superintegrable systems

Tremblay, Frédérick, Turbiner, Alexander V. et Winternitz, Pavel, Periodic orbits for an infinite family of classical superintegrable systems 43, 015202, 14 (2010), , J. Phys. A

#### Invariant difference schemes and their application to ${\rm sl}(2,\Bbb R)$ invariant ordinary differential equations

Rebelo, R. et Winternitz, P., Invariant difference schemes and their application to ${\rm sl}(2,\Bbb R)$ invariant ordinary differential equations 42, 454016, 10 (2009), , J. Phys. A

#### Integrable and superintegrable systems with spin in three-dimensional Euclidean space

Winternitz, Pavel et Yurdu\c sen, \. Ismet, Integrable and superintegrable systems with spin in three-dimensional Euclidean space 42, 385203, 20 (2009), , J. Phys. A

#### An infinite family of solvable and integrable quantum systems on a plane

Tremblay, Frédérick, Turbiner, Alexander V. et Winternitz, Pavel, An infinite family of solvable and integrable quantum systems on a plane 42, 242001, 10 (2009), , J. Phys. A

#### All solvable extensions of a class of nilpotent Lie algebras of dimension $n$ and degree of nilpotency $n-1$

\v Snobl, L. et Winternitz, P., All solvable extensions of a class of nilpotent Lie algebras of dimension $n$ and degree of nilpotency $n-1$ 42, 105201, 16 (2009), , J. Phys. A

#### Reduction of superintegrable systems: the anisotropic harmonic oscillator

Rodríguez, Miguel A., Tempesta, Piergiulio et Winternitz, Pavel, Reduction of superintegrable systems: the anisotropic harmonic oscillator 78, 046608, 6 (2008), , Phys. Rev. E (3)

#### Symmetry preserving discretization of ${\rm SL}(2,\Bbb R)$ invariant equations

Bourlioux, Anne, Rebelo, Raphaël et Winternitz, Pavel, Symmetry preserving discretization of ${\rm SL}(2,\Bbb R)$ invariant equations 15, 362--372 (2008), , J. Nonlinear Math. Phys.

#### Superintegrable systems with third-order integrals of motion

Marquette, Ian et Winternitz, Pavel, Superintegrable systems with third-order integrals of motion 41, 304031, 10 (2008), , J. Phys. A

#### Heisenberg algebra, umbral calculus and orthogonal polynomials

Dattoli, G., Levi, D. et Winternitz, P., Heisenberg algebra, umbral calculus and orthogonal polynomials 49, 053509, 19 (2008), , J. Math. Phys.

#### Erratum: Polynomial Poisson algebras for classical superintegrable systems with a third order integral of motion'' [J. Math. Phys. {\bf 48} (2007), no. 1, 012902, 16 pp.; MR2292612]

Marquette, I. et Winternitz, P., Erratum: Polynomial Poisson algebras for classical superintegrable systems with a third order integral of motion'' [J. Math. Phys. {\bf 48} (2007), no. 1, 012902, 16 pp.; MR2292612] 49, 019901, 1 (2008), , J. Math. Phys.

#### Erratum: A class of superintegrable systems of Calogero type'' [J. Math. Phys. {\bf 47} (2006), no. 9, 093505, 8 pp.; èfcno 2263659]

Smirnov, Roman G. et Winternitz, Pavel, Erratum: A class of superintegrable systems of Calogero type'' [J. Math. Phys. {\bf 47} (2006), no. 9, 093505, 8 pp.; èfcno 2263659] 48, 079902, 1 (2007), , J. Math. Phys.

#### Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Marquette, Ian et Winternitz, Pavel, Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion 48, 012902, 16 (2007), , J. Math. Phys.

#### Quasiseparation of variables in the Schrödinger equation with a magnetic field

Charest, F., Hudon, C. et Winternitz, P., Quasiseparation of variables in the Schrödinger equation with a magnetic field 48, 012105, 16 (2007), , J. Math. Phys.

#### Integrable and superintegrable systems with spin

Winternitz, Pavel et Yurdu\c sen, \. Ismet, Integrable and superintegrable systems with spin 47, 103509, 10 (2006), , J. Math. Phys.

#### A class of superintegrable systems of Calogero type

Smirnov, Roman G. et Winternitz, Pavel, A class of superintegrable systems of Calogero type 47, 093505, 8 (2006), , J. Math. Phys.

#### Difference schemes with point symmetries and their numerical tests

Bourlioux, A., Cyr-Gagnon, C. et Winternitz, P., Difference schemes with point symmetries and their numerical tests 39, 6877--6896 (2006), , J. Phys. A

#### Continuous symmetries of difference equations

Levi, Decio et Winternitz, Pavel, Continuous symmetries of difference equations 39, R1--R63 (2006), , J. Phys. A

#### Discretization of partial differential equations preserving their physical symmetries

Valiquette, F. et Winternitz, P., Discretization of partial differential equations preserving their physical symmetries 38, 9765--9783 (2005), , J. Phys. A

#### Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and Schrödinger equations

Kalnins, E. G., Thomova, Z. et Winternitz, P., Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and Schrödinger equations 12, 178--208 (2005), , J. Nonlinear Math. Phys.

#### A class of solvable Lie algebras and their Casimir invariants

\v Snobl, L. et Winternitz, P., A class of solvable Lie algebras and their Casimir invariants 38, 2687--2700 (2005), , J. Phys. A

#### Umbral calculus, difference equations and the discrete Schrödinger equation

Levi, Decio, Tempesta, Piergiulio et Winternitz, Pavel, Umbral calculus, difference equations and the discrete Schrödinger equation 45, 4077--4105 (2004), , J. Math. Phys.

#### Lie symmetries and exact solutions of first-order difference schemes

Rodríguez, M. A. et Winternitz, P., Lie symmetries and exact solutions of first-order difference schemes 37, 6129--6142 (2004), , J. Phys. A

#### Equivalence classes and symmetries of the variable coefficient Kadomtsev-Petviashvili equation

Güngör, F. et Winternitz, P., Equivalence classes and symmetries of the variable coefficient Kadomtsev-Petviashvili equation 35, 381--396 (2004), , Nonlinear Dynam.

#### Discrete matrix Riccati equations with superposition formulas

Penskoi, Alexei V. et Winternitz, Pavel, Discrete matrix Riccati equations with superposition formulas 294, 533--547 (2004), , J. Math. Anal. Appl.

#### Integrable and superintegrable quantum systems in a magnetic field

Bérubé, Josée et Winternitz, Pavel, Integrable and superintegrable quantum systems in a magnetic field 45, 1959--1973 (2004), , J. Math. Phys.

#### Continuous symmetries of Lagrangians and exact solutions of discrete equations

Dorodnitsyn, Vladimir, Kozlov, Roman et Winternitz, Pavel, Continuous symmetries of Lagrangians and exact solutions of discrete equations 45, 336--359 (2004), , J. Math. Phys.

#### Symmetries, Lagrangian formalism and integration of second order ordinary difference equations

Dorodnitsyn, Vladimir, Kozlov, Roman et Winternitz, Pavel, Symmetries, Lagrangian formalism and integration of second order ordinary difference equations 10, 41--56 (2003), , J. Nonlinear Math. Phys.

#### Superintegrable systems in Darboux spaces

Kalnins, E. G., Kress, J. M., Miller, Jr., W. et Winternitz, P., Superintegrable systems in Darboux spaces 44, 5811--5848 (2003), , J. Math. Phys.

#### Weak transversality and partially invariant solutions

Grundland, A. M., Tempesta, P. et Winternitz, P., Weak transversality and partially invariant solutions 44, 2704--2722 (2003), , J. Math. Phys.

#### Lie symmetries and superintegrability in quantum mechanics

Sheftel, M. B., Tempesta, P. et Winternitz, P., Lie symmetries and superintegrability in quantum mechanics 65, 1144--1148 (2002), , Phys. Atomic Nuclei

#### A superintegrable time-dependent system with Kac-Moody symmetry

Daboul, J. et Winternitz, P., A superintegrable time-dependent system with Kac-Moody symmetry 65, 1000--1007 (2002), , Phys. Atomic Nuclei

#### Generalized Kadomtsev-Petviashvili equation with an infinite-dimensional symmetry algebra

Güngör, F. et Winternitz, P., Generalized Kadomtsev-Petviashvili equation with an infinite-dimensional symmetry algebra 276, 314--328 (2002), , J. Math. Anal. Appl.

#### Superintegrability with third-order integrals in quantum and classical mechanics

Gravel, Simon et Winternitz, Pavel, Superintegrability with third-order integrals in quantum and classical mechanics 43, 5902--5912 (2002), , J. Math. Phys.

#### Lie point symmetries and commuting flows for equations on lattices

Levi, D. et Winternitz, P., Lie point symmetries and commuting flows for equations on lattices 35, 2249--2262 (2002), , J. Phys. A

#### Separation of variables and subgroup bases on $n$-dimensional hyperboloids

Pogosyan, G. S. et Winternitz, P., Separation of variables and subgroup bases on $n$-dimensional hyperboloids 43, 3387--3410 (2002), , J. Math. Phys.

#### Quantum superintegrability and exact solvability in $n$ dimensions

Rodriguez, Miguel A. et Winternitz, Pavel, Quantum superintegrability and exact solvability in $n$ dimensions 43, 1309--1322 (2002), , J. Math. Phys.

#### Superintegrability in a two-dimensional space of nonconstant curvature

Kalnins, E. G., Kress, J. M. et Winternitz, P., Superintegrability in a two-dimensional space of nonconstant curvature 43, 970--983 (2002), , J. Math. Phys.

#### Derivation of Graf's addition theorem by contractions of the group $\rm SO(3)$

Vinternits, P., Volf, K. B., Pogosyan, G. S. et Sisakyan, A. N., Derivation of Graf's addition theorem by contractions of the group $\rm SO(3)$ 129, 227--229 (2001), , Teoret. Mat. Fiz.

#### Lie symmetries of multidimensional difference equations

Levi, D., Tremblay, S. et Winternitz, P., Lie symmetries of multidimensional difference equations 34, 9507--9524 (2001), , J. Phys. A

#### Invariants of the nilpotent and solvable triangular Lie algebras

Tremblay, S. et Winternitz, P., Invariants of the nilpotent and solvable triangular Lie algebras 34, 9085--9099 (2001), , J. Phys. A

#### Symmetries of the discrete nonlinear Schrödinger equation

Èrnandes Eredero, R., Levi, D. et Vinternits, P., Symmetries of the discrete nonlinear Schrödinger equation 127, 379--387 (2001), , Teoret. Mat. Fiz.

#### Group foliation and non-invariant solutions of the heavenly equation

Martina, L., Sheftel, M. B. et Winternitz, P., Group foliation and non-invariant solutions of the heavenly equation 34, 9243--9263 (2001), , J. Phys. A

#### Symmetry classification of diatomic molecular chains

Lafortune, S., Tremblay, S. et Winternitz, P., Symmetry classification of diatomic molecular chains 42, 5341--5357 (2001), , J. Math. Phys.

#### Exact solvability of superintegrable systems

Tempesta, Piergiulio, Turbiner, Alexander V. et Winternitz, Pavel, Exact solvability of superintegrable systems 42, 4248--4257 (2001), , J. Math. Phys.

#### Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

Sheftel, M. B., Tempesta, P. et Winternitz, P., Recursion operators, higher-order symmetries and superintegrability in quantum mechanics 51, 392--399 (2001), , Czechoslovak J. Phys.

#### Continuous symmetries of equations on lattices

Levi, D., Tremblay, S. et Winternitz, P., Continuous symmetries of equations on lattices 51, 349--356 (2001), , Czechoslovak J. Phys.

#### Relation between Bäcklund transformations and higher continuous symmetries of the Toda equation

Hernàndez Heredero, R., Levi, D., Rodríguez, M. A. et Winternitz, P., Relation between Bäcklund transformations and higher continuous symmetries of the Toda equation 34, 2459--2465 (2001), , J. Phys. A

#### Time-dependent realization of the infinite-dimensional hydrogen algebra

Daboul, Jamil et Winternitz, Pavel, Time-dependent realization of the infinite-dimensional hydrogen algebra 282, 163--168 (2001), , Phys. Lett. A

#### Contractions of Lie algebras and the separation of variables: interbase expansions

Izmestev, A. A., Pogosyan, G. S., Sissakian, A. N. et Winternitz, P., Contractions of Lie algebras and the separation of variables: interbase expansions 34, 521--554 (2001), , J. Phys. A

#### Superintegrable systems in quantum mechanics and classical Lie theory

Sheftel, M. B., Tempesta, P. et Winternitz, P., Superintegrable systems in quantum mechanics and classical Lie theory 42, 659--673 (2001), , J. Math. Phys.

#### Lie point symmetries of difference equations and lattices

Levi, D., Tremblay, S. et Winternitz, P., Lie point symmetries of difference equations and lattices 33, 8507--8523 (2000), , J. Phys. A

#### Invariant solutions of hydrodynamic-type equations

Grundland, A. M., Sheftel, M. B. et Winternitz, P., Invariant solutions of hydrodynamic-type equations 33, 8193--8215 (2000), , J. Phys. A

#### Point symmetries of generalized Toda field theories. II. Symmetry reduction

Martina, L., Lafortune, S. et Winternitz, P., Point symmetries of generalized Toda field theories. II. Symmetry reduction 33, 6431--6446 (2000), , J. Phys. A

#### Bases for representations of quantum algebras

Atakishiyev, N. M. et Winternitz, P., Bases for representations of quantum algebras 33, 5303--5313 (2000), , J. Phys. A

#### Lie point symmetry preserving discretizations for variable coefficient Korteweg-de Vries equations

Dorodnitsyn, V. et Winternitz, P., Lie point symmetry preserving discretizations for variable coefficient Korteweg-de Vries equations 22, 49--59 (2000), , Nonlinear Dynam.

#### Lie algebra contractions and symmetries of the Toda hierarchy

Hernàndez Heredero, R., Levi, D., Rodríguez, M. A. et Winternitz, P., Lie algebra contractions and symmetries of the Toda hierarchy 33, 5025--5040 (2000), , J. Phys. A

#### Discrete systems related to some equations of the Painlevé-Gambier classification

Lafortune, S., Grammaticos, B., Ramani, A. et Winternitz, P., Discrete systems related to some equations of the Painlevé-Gambier classification 270, 55--61 (2000), , Phys. Lett. A

#### Huygens' principle and separation of variables

Berest, Yuri et Winternitz, Pavel, Huygens' principle and separation of variables 12, 159--180 (2000), , Rev. Math. Phys.

#### Integrable and superintegrable Hamiltonian systems in magnetic fields

McSween, Eric et Winternitz, Pavel, Integrable and superintegrable Hamiltonian systems in magnetic fields 41, 2957--2967 (2000), , J. Math. Phys.

#### Point symmetries of generalized Toda field theories

Lafortune, S., Winternitz, P. et Martina, L., Point symmetries of generalized Toda field theories 33, 2419--2435 (2000), , J. Phys. A

#### Lie group classification of second-order ordinary difference equations

Dorodnitsyn, Vladimir, Kozlov, Roman et Winternitz, Pavel, Lie group classification of second-order ordinary difference equations 41, 480--504 (2000), , J. Math. Phys.

#### Solutions of nonlinear differential and difference equations with superposition formulas

Turbiner, Alexander et Winternitz, Pavel, Solutions of nonlinear differential and difference equations with superposition formulas 50, 189--201 (1999), , Lett. Math. Phys.

#### Singularity analysis, balance equations and soliton solution of the nonlocal complex Ginzburg-Landau equation

Ankiewicz, A., Akhmediev, N. N. et Winternitz, P., Singularity analysis, balance equations and soliton solution of the nonlocal complex Ginzburg-Landau equation 36, 11--24 (1999), , J. Engrg. Math.

#### Nonlinear superposition formulas based on imprimitive group action

Havlí\v cek, M., Po?ta, S. et Winternitz, P., Nonlinear superposition formulas based on imprimitive group action 40, 3104--3122 (1999), , J. Math. Phys.

#### Symmetries of discrete dynamical systems involving two species

Gómez-Ullate, D., Lafortune, S. et Winternitz, P., Symmetries of discrete dynamical systems involving two species 40, 2782--2804 (1999), , J. Math. Phys.

#### Symmetries of the discrete Burgers equation

Hernàndez Heredero, R., Levi, D. et Winternitz, P., Symmetries of the discrete Burgers equation 32, 2685--2695 (1999), , J. Phys. A

#### Maximal abelian subalgebras of $e(p,q)$ algebras

Thomova, Z. et Winternitz, P., Maximal abelian subalgebras of $e(p,q)$ algebras 291, 245--274 (1999), , Linear Algebra Appl.

#### Group invariant solutions for the $N=2$ super Korteweg-de-Vries equation

Ayari, M. A., Hussin, V. et Winternitz, P., Group invariant solutions for the $N=2$ super Korteweg-de-Vries equation 40, 1951--1965 (1999), , J. Math. Phys.

#### Contractions of Lie algebras and separation of variables. The $n$-dimensional sphere

Izmestev, A. A., Pogosyan, G. S., Sissakian, A. N. et Winternitz, P., Contractions of Lie algebras and separation of variables. The $n$-dimensional sphere 40, 1549--1573 (1999), , J. Math. Phys.

#### Graded contractions of the Lie algebra $e(2,1)$

Patera, J., Pogosyan, G. et Winternitz, P., Graded contractions of the Lie algebra $e(2,1)$ 32, 805--826 (1999), , J. Phys. A

#### Lie group contractions and separation of variables

Winternitz, P., Lie group contractions and separation of variables 61, 1705--1712 (1998), , Phys. Atomic Nuclei

#### Discretizing families of linearizable equations

Grammaticos, B., Ramani, A. et Winternitz, P., Discretizing families of linearizable equations 245, 382--388 (1998), , Phys. Lett. A

#### Solvable Lie algebras with triangular nilradicals

Tremblay, S. et Winternitz, P., Solvable Lie algebras with triangular nilradicals 31, 789--806 (1998), , J. Phys. A

#### Solutions of $(2+1)$-dimensional spin systems

Thomova, Z., Winternitz, P. et Zakrzewski, W. J., Solutions of $(2+1)$-dimensional spin systems 39, 3927--3944 (1998), , J. Math. Phys.

#### Maximal abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces

Thomova, Z. et Winternitz, P., Maximal abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces 31, 1831--1858 (1998), , J. Phys. A

#### $P_\infty$ algebra of KP, free fermions and $2$-cocycle in the Lie algebra of pseudodifferential operators

Orlov, A. Yu. et Winternitz, P., $P_\infty$ algebra of KP, free fermions and $2$-cocycle in the Lie algebra of pseudodifferential operators 11, 3159--3193 (1997), , Internat. J. Modern Phys. B

#### $P_\infty$-algebra of symmetries of the Kadomtsev-Petviashvili equations, free fermions and $2$-cocycles in the Lie algebra of pseudodifferential operators

Winternitz, P. et Orlov, A. Yu., $P_\infty$-algebra of symmetries of the Kadomtsev-Petviashvili equations, free fermions and $2$-cocycles in the Lie algebra of pseudodifferential operators 113, 231--260 (1997), , Teoret. Mat. Fiz.

#### Lie-theoretical generalization and discretization of the Pinney equation

Rogers, C., Schief, W. K. et Winternitz, P., Lie-theoretical generalization and discretization of the Pinney equation 216, 246--264 (1997), , J. Math. Anal. Appl.

#### Symmetries and solutions of the vector nonlinear Schrödinger equation

Sciarrino, A. et Winternitz, P., Symmetries and solutions of the vector nonlinear Schrödinger equation 112, 853--871 (1997), , Nuovo Cimento Soc. Ital. Fis. B (12)

#### Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev-Petviashvili hierarchy

Orlov, A. Yu. et Winternitz, P., Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev-Petviashvili hierarchy 38, 4644--4674 (1997), , J. Math. Phys.

#### Successive refinements of gradings and graded contractions of ${\rm sl}(3,{\bf C})$

Winternitz, P., Successive refinements of gradings and graded contractions of ${\rm sl}(3,{\bf C})$ 12, 109--115 (1997), , Internat. J. Modern Phys. A

#### Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid

Izmestev, A. A., Pogosyan, G. S., Sissakian, A. N. et Winternitz, P., Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid 12, 53--61 (1997), , Internat. J. Modern Phys. A

#### Lie group formalism for difference equations

Levi, D., Vinet, L. et Winternitz, P., Lie group formalism for difference equations 30, 633--649 (1997), , J. Phys. A

#### Grading refinements in the contractions of Lie algebras and their invariants

Ait Abdelmalek, M., Leng, X., Patera, J. et Winternitz, P., Grading refinements in the contractions of Lie algebras and their invariants 29, 7519--7543 (1996), , J. Phys. A

#### Contractions of Lie algebras and separation of variables

Izmestev, A. A., Pogosyan, G. S., Sissakian, A. N. et Winternitz, P., Contractions of Lie algebras and separation of variables 29, 5949--5962 (1996), , J. Phys. A

#### Symmetries of discrete dynamical systems

Levi, D. et Winternitz, P., Symmetries of discrete dynamical systems 37, 5551--5576 (1996), , J. Math. Phys.

#### The conformal group ${\rm SU}(2,2)$ and integrable systems on a Lorentzian hyperboloid

del Olmo, M. A., Rodríguez, M. A. et Winternitz, P., The conformal group ${\rm SU}(2,2)$ and integrable systems on a Lorentzian hyperboloid 44, 199--233 (1996), , Fortschr. Phys.

#### Superposition formulas for pseudounitary matrix Riccati equations

Lafortune, Stéphane et Winternitz, Pavel, Superposition formulas for pseudounitary matrix Riccati equations 37, 1539--1550 (1996), , J. Math. Phys.

#### On the solutions of the ${\bf C}{\rm P}^1$ model in $(2+1)$ dimensions

Grundland, A. M., Winternitz, P. et Zakrzewski, W. J., On the solutions of the ${\bf C}{\rm P}^1$ model in $(2+1)$ dimensions 37, 1501--1520 (1996), , J. Math. Phys.

#### Classical and quantum integrable systems in $\widetilde{\germ g\germ l}(2)^{+*}$ and separation of variables

Harnad, J. et Winternitz, P., Classical and quantum integrable systems in $\widetilde{\germ g\germ l}(2)^{+*}$ and separation of variables 172, 263--285 (1995), , Comm. Math. Phys.

#### Harmonics on hyperspheres, separation of variables and the Bethe ansatz

Harnad, J. et Winternitz, P., Harmonics on hyperspheres, separation of variables and the Bethe ansatz 33, 61--74 (1995), , Lett. Math. Phys.

#### Maximal abelian subalgebras of complex Euclidean Lie algebras

Kalnins, E. G. et Winternitz, P., Maximal abelian subalgebras of complex Euclidean Lie algebras 72, 389--404 (1994), , Canad. J. Phys.

#### Generalized Casimir operators of solvable Lie algebras with abelian nilradicals

Ndogmo, J. C. et Winternitz, P., Generalized Casimir operators of solvable Lie algebras with abelian nilradicals 27, 2787--2800 (1994), , J. Phys. A

#### Solvable Lie algebras with abelian nilradicals

Ndogmo, J. C. et Winternitz, P., Solvable Lie algebras with abelian nilradicals 27, 405--423 (1994), , J. Phys. A

#### Symmetries and conditional symmetries of a nonrelativistic Chern-Simons system

Levi, Decio, Vinet, Luc et Winternitz, Pavel, Symmetries and conditional symmetries of a nonrelativistic Chern-Simons system 230, 101--117 (1994), , Ann. Physics

#### Symmetry analysis of the Infeld-Rowlands equation

Faucher, M. et Winternitz, P., Symmetry analysis of the Infeld-Rowlands equation 48, 3066--3071 (1993), , Phys. Rev. E (3)

#### Symmetry classes of variable coefficient nonlinear Schrödinger equations

Gagnon, L. et Winternitz, P., Symmetry classes of variable coefficient nonlinear Schrödinger equations 26, 7061--7076 (1993), , J. Phys. A

#### Representations of the quantum algebra ${\rm su}_q(2)$ on a real two-dimensional sphere

Rideau, G. et Winternitz, P., Representations of the quantum algebra ${\rm su}_q(2)$ on a real two-dimensional sphere 34, 6030--6044 (1993), , J. Math. Phys.

#### On the relation between weak and strong invariance of differential equations

Cariñena, José F., del Olmo, M. A. et Winternitz, P., On the relation between weak and strong invariance of differential equations 29, 151--163 (1993), , Lett. Math. Phys.

#### Integrable systems based on ${\rm SU}(p,q)$ homogeneous manifolds

del Olmo, M. A., Rodriguez, M. A. et Winternitz, P., Integrable systems based on ${\rm SU}(p,q)$ homogeneous manifolds 34, 5118--5139 (1993), , J. Math. Phys.

#### Symmetries and conditional symmetries of differential-difference equations

Levi, D. et Winternitz, P., Symmetries and conditional symmetries of differential-difference equations 34, 3713--3730 (1993), , J. Math. Phys.

#### On the symmetry groups of the intrinsic generalized wave and sine-Gordon equations

Tenenblat, K. et Winternitz, P., On the symmetry groups of the intrinsic generalized wave and sine-Gordon equations 34, 3527--3542 (1993), , J. Math. Phys.

#### Group theory and solutions of classical field theories with polynomial nonlinearities

Grundland, A. M., Tuszy\'nski, J. A. et Winternitz, P., Group theory and solutions of classical field theories with polynomial nonlinearities 23, 633--665 (1993), , Found. Phys.

#### Solvable Lie algebras with Heisenberg ideals

Rubin, J. L. et Winternitz, P., Solvable Lie algebras with Heisenberg ideals 26, 1123--1138 (1993), , J. Phys. A

#### Evolution equations invariant under two-dimensional space-time Schrödinger group

Rideau, G. et Winternitz, P., Evolution equations invariant under two-dimensional space-time Schrödinger group 34, 558--570 (1993), , J. Math. Phys.

#### Symmetries of variable coefficient Korteweg-de-Vries equations

Gazeau, J.-P. et Winternitz, P., Symmetries of variable coefficient Korteweg-de-Vries equations 33, 4087--4102 (1992), , J. Math. Phys.

#### Partially invariant solutions of a class of nonlinear Schrödinger equations

Martina, L., Soliani, G. et Winternitz, P., Partially invariant solutions of a class of nonlinear Schrödinger equations 25, 4425--4435 (1992), , J. Phys. A

#### Allowed transformations and symmetry classes of variable coefficient Korteweg-de-Vries equations

Winternitz, Pavel et Gazeau, J.-P., Allowed transformations and symmetry classes of variable coefficient Korteweg-de-Vries equations 167, 246--250 (1992), , Phys. Lett. A

#### Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations

Martina, L. et Winternitz, P., Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations 33, 2718--2727 (1992), , J. Math. Phys.

#### Maximal abelian subalgebras of pseudo-orthogonal Lie algebras

Hussin, V., Winternitz, P. et Zassenhaus, H., Maximal abelian subalgebras of pseudo-orthogonal Lie algebras 173, 125--163 (1992), , Linear Algebra Appl.

#### Exact solutions of the stimulated-Raman-scattering equations

Levi, D., Menyuk, C. R. et Winternitz, P., Exact solutions of the stimulated-Raman-scattering equations 44, 6057--6070 (1991), , Phys. Rev. A (3)

#### Symmetry properties and solutions of nonlinear dispersive thin-film equations in three dimensions

Melkonian, S. et Winternitz, P., Symmetry properties and solutions of nonlinear dispersive thin-film equations in three dimensions 32, 3213--3222 (1991), , J. Math. Phys.

#### The computer calculation of Lie point symmetries of large systems of differential equations

Champagne, B., Hereman, W. et Winternitz, P., The computer calculation of Lie point symmetries of large systems of differential equations 66, 319--340 (1991), , Comput. Phys. Comm.

#### Graded contractions of ${\rm sl}(3,{\bf C})$

Couture, M., Patera, J., Sharp, R. T. et Winternitz, P., Graded contractions of ${\rm sl}(3,{\bf C})$ 32, 2310--2318 (1991), , J. Math. Phys.

#### Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation

Clarkson, P. A. et Winternitz, P., Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation 49, 257--272 (1991), , Phys. D

#### Continuous symmetries of discrete equations

Levi, D. et Winternitz, P., Continuous symmetries of discrete equations 152, 335--338 (1991), , Phys. Lett. A

#### Group theoretical analysis of dispersive long wave equations in two space dimensions

Paquin, G. et Winternitz, P., Group theoretical analysis of dispersive long wave equations in two space dimensions 46, 122--138 (1990), , Phys. D

#### Maximal abelian subalgebras of complex orthogonal Lie algebras

Hussin, V., Winternitz, P. et Zassenhaus, H., Maximal abelian subalgebras of complex orthogonal Lie algebras 141, 183--220 (1990), , Linear Algebra Appl.

#### Periodicity and quasi-periodicity for super-integrable Hamiltonian systems

Kibler, M. et Winternitz, P., Periodicity and quasi-periodicity for super-integrable Hamiltonian systems 147, 338--342 (1990), , Phys. Lett. A

#### Superposition formulas for nonlinear superequations

Beckers, J., Gagnon, L., Hussin, V. et Winternitz, P., Superposition formulas for nonlinear superequations 31, 2528--2534 (1990), , J. Math. Phys.

#### Point symmetries of conditionally integrable nonlinear evolution equations

Rubin, J. et Winternitz, P., Point symmetries of conditionally integrable nonlinear evolution equations 31, 2085--2090 (1990), , J. Math. Phys.

#### Maximal abelian subalgebras of pseudounitary Lie algebras

del Olmo, M. A., Rodríguez, M. A., Winternitz, P. et Zassenhaus, H., Maximal abelian subalgebras of pseudounitary Lie algebras 135, 79--151 (1990), , Linear Algebra Appl.

#### Nonlinear equations invariant under the Poincaré, similitude, and conformal groups in two-dimensional space-time

Rideau, G. et Winternitz, P., Nonlinear equations invariant under the Poincaré, similitude, and conformal groups in two-dimensional space-time 31, 1095--1105 (1990), , J. Math. Phys.

#### Symmetries of semiclassical gravity in two dimensions

Floreanini, Roberto, Lina, Jean-Marc, Vinet, Luc et Winternitz, Pavel, Symmetries of semiclassical gravity in two dimensions 41, 1862--1866 (1990), , Phys. Rev. D (3)

#### Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem

Martina, L. et Winternitz, P., Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem 196, 231--277 (1989), , Ann. Physics

#### Group theoretical analysis of a rotating shallow liquid in a rigid container

Levi, D., Nucci, M. C., Rogers, C. et Winternitz, P., Group theoretical analysis of a rotating shallow liquid in a rigid container 22, 4743--4767 (1989), , J. Phys. A

#### Nonclassical symmetry reduction: example of the Boussinesq equation

Levi, D. et Winternitz, P., Nonclassical symmetry reduction: example of the Boussinesq equation 22, 2915--2924 (1989), , J. Phys. A

#### Solitons in shallow seas of variable depth and in marine straits

David, D., Levi, D. et Winternitz, P., Solitons in shallow seas of variable depth and in marine straits 80, 1--23 (1989), , Stud. Appl. Math.

#### Symmetries of the self-dual Einstein equations. I. The infinite-dimensional symmetry group and its low-dimensional subgroups

Boyer, C. P. et Winternitz, P., Symmetries of the self-dual Einstein equations. I. The infinite-dimensional symmetry group and its low-dimensional subgroups 30, 1081--1094 (1989), , J. Math. Phys.

#### Lie symmetries of a generalised nonlinear Schrödinger equation. III. Reductions to third-order ordinary differential equations

Gagnon, L., Grammaticos, B., Ramani, A. et Winternitz, P., Lie symmetries of a generalised nonlinear Schrödinger equation. III. Reductions to third-order ordinary differential equations 22, 499--509 (1989), , J. Phys. A

#### Lie symmetries of a generalised nonlinear Schrödinger equation. II. Exact solutions

Gagnon, L. et Winternitz, P., Lie symmetries of a generalised nonlinear Schrödinger equation. II. Exact solutions 22, 469--497 (1989), , J. Phys. A

#### Exact solutions of the cubic and quintic nonlinear Schrödinger equation for a cylindrical geometry

Gagnon, L. et Winternitz, P., Exact solutions of the cubic and quintic nonlinear Schrödinger equation for a cylindrical geometry 39, 296--306 (1989), , Phys. Rev. A (3)

#### Exact solutions of the spherical quintic nonlinear Schrödinger equation

Gagnon, L. et Winternitz, P., Exact solutions of the spherical quintic nonlinear Schrödinger equation 134, 276--281 (1989), , Phys. Lett. A

#### Nonlinear equations with superposition formulas and exceptional group $G_2$. III. The superposition formulas

Gagnon, L., Hussin, V. et Winternitz, P., Nonlinear equations with superposition formulas and exceptional group $G_2$. III. The superposition formulas 29, 2145--2155 (1988), , J. Math. Phys.

#### On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, Levi decompositions, and nilradicals

Rand, D., Winternitz, P. et Zassenhaus, H., On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, Levi decompositions, and nilradicals 109, 197--246 (1988), , Linear Algebra Appl.

#### Lie algebras under constraints and nonbijective canonical transformations

Kibler, Maurice et Winternitz, Pavel, Lie algebras under constraints and nonbijective canonical transformations 21, 1787--1803 (1988), , J. Phys. A

#### Lie symmetries of a generalised nonlinear Schrödinger equation. I. The symmetry group and its subgroups

Gagnon, L. et Winternitz, P., Lie symmetries of a generalised nonlinear Schrödinger equation. I. The symmetry group and its subgroups 21, 1493--1511 (1988), , J. Phys. A

#### The cylindrical Kadomtsev-Petviashvili equation: its Kac-Moody-Virasoro algebra and relation to KP equation

Levi, D. et Winternitz, P., The cylindrical Kadomtsev-Petviashvili equation: its Kac-Moody-Virasoro algebra and relation to KP equation 129, 165--167 (1988), , Phys. Lett. A

#### Equations invariant under the symmetry group of the Kadomtsev-Petviashvili equation

David, D., Levi, D. et Winternitz, P., Equations invariant under the symmetry group of the Kadomtsev-Petviashvili equation 129, 161--164 (1988), , Phys. Lett. A

#### The classification of complete sets of operators commuting with the Dirac operator in Minkowski space-time

Kamran, N., Légaré, M., McLenaghan, R. G. et Winternitz, P., The classification of complete sets of operators commuting with the Dirac operator in Minkowski space-time 29, 403--411 (1988), , J. Math. Phys.

#### On the infinite-dimensional symmetry group of the Davey-Stewartson equations

Champagne, B. et Winternitz, P., On the infinite-dimensional symmetry group of the Davey-Stewartson equations 29, 1--8 (1988), , J. Math. Phys.

#### Integrable nonlinear equations for water waves in straits of varying depth and width

David, D., Levi, D. et Winternitz, P., Integrable nonlinear equations for water waves in straits of varying depth and width 76, 133--168 (1987), , Stud. Appl. Math.

#### Dynamical invariance algebra of the Hartmann potential

Kibler, M. et Winternitz, P., Dynamical invariance algebra of the Hartmann potential 20, 4097--4108 (1987), , J. Phys. A

#### PASCAL programs for the identification of Lie algebras. II. SPLIT---a program to decompose parameter-free and parameter-dependent Lie algebras into direct sums

Rand, D. W., Winternitz, P. et Zassenhaus, H., PASCAL programs for the identification of Lie algebras. II. SPLIT---a program to decompose parameter-free and parameter-dependent Lie algebras into direct sums 46, 297--309 (1987), , Comput. Phys. Comm.

#### Exact solutions of the multidimensional classical $\phi^6$-field equations obtained by symmetry reduction

Winternitz, P., Grundland, A. M. et Tuszy\'nski, J. A., Exact solutions of the multidimensional classical $\phi^6$-field equations obtained by symmetry reduction 28, 2194--2212 (1987), , J. Math. Phys.

#### Nonlinear differential equations and Lie superalgebras

Beckers, J., Gagnon, L., Hussin, V. et Winternitz, P., Nonlinear differential equations and Lie superalgebras 13, 113--120 (1987), , Lett. Math. Phys.

#### Superposition formulas for the rectangular matrix Riccati equations

del Olmo, M. A., Rodríguez, M. A. et Winternitz, P., Superposition formulas for the rectangular matrix Riccati equations 28, 530--535 (1987), , J. Math. Phys.

#### Nonlinear equations with superposition formulas and the exceptional group $G_2$. II. Classification of the equations

Beckers, J., Hussin, V. et Winternitz, P., Nonlinear equations with superposition formulas and the exceptional group $G_2$. II. Classification of the equations 28, 520--529 (1987), , J. Math. Phys.

#### New classes of exact solutions of the $\phi^6$ model in $3+1$ dimensions

Grundland, A. M., Tuszy\'nski, J. A. et Winternitz, P., New classes of exact solutions of the $\phi^6$ model in $3+1$ dimensions 119, 340--344 (1987), , Phys. Lett. A

#### ODEPAINLEVE---a MACSYMA package for Painlevé analysis of ordinary differential equations

Rand, D. W. et Winternitz, P., ODEPAINLEVE---a MACSYMA package for Painlevé analysis of ordinary differential equations 42, 359--383 (1986), , Comput. Phys. Comm.

#### Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation

David, D., Levi, D. et Winternitz, P., Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation 118, 390--394 (1986), , Phys. Lett. A

#### Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable?

Dorizzi, B., Grammaticos, B., Ramani, A. et Winternitz, P., Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? 27, 2848--2852 (1986), , J. Math. Phys.

#### Nonlinear equations with superposition formulas and the exceptional group $G_2$. I. Complex and real forms of $g_2$ and their maximal subalgebras

Beckers, J., Hussin, V. et Winternitz, P., Nonlinear equations with superposition formulas and the exceptional group $G_2$. I. Complex and real forms of $g_2$ and their maximal subalgebras 27, 2217--2227 (1986), , J. Math. Phys.

#### On the integrability and perturbations of systems of ODEs with nonlinear superposition principles

Bountis, T. C., Papageorgiou, V. et Winternitz, P., On the integrability and perturbations of systems of ODEs with nonlinear superposition principles 18, 211--212 (1986), , Phys. D

#### Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra

David, D., Kamran, N., Levi, D. et Winternitz, P., Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra 27, 1225--1237 (1986), , J. Math. Phys.

#### On the integrability of systems of nonlinear ordinary differential equations with superposition principles

Bountis, T. C., Papageorgiou, V. et Winternitz, P., On the integrability of systems of nonlinear ordinary differential equations with superposition principles 27, 1215--1224 (1986), , J. Math. Phys.

#### Complex parabolic subgroups of $G_2$ and nonlinear differential equations

Beckers, J., Hussin, V. et Winternitz, P., Complex parabolic subgroups of $G_2$ and nonlinear differential equations 11, 81--86 (1986), , Lett. Math. Phys.

#### Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles

del Olmo, M. A., Rodríguez, M. A. et Winternitz, P., Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles 27, 14--23 (1986), , J. Math. Phys.

#### Integrable Hamiltonian systems with velocity-dependent potentials

Dorizzi, B., Grammaticos, B., Ramani, A. et Winternitz, P., Integrable Hamiltonian systems with velocity-dependent potentials 26, 3070--3079 (1985), , J. Math. Phys.

#### Subalgebras of loop algebras and symmetries of the Kadomtsev-Petviashvili equation

David, D., Kamran, N., Levi, D. et Winternitz, P., Subalgebras of loop algebras and symmetries of the Kadomtsev-Petviashvili equation 55, 2111--2113 (1985), , Phys. Rev. Lett.

#### Abelian integrals and the reduction method for an integrable Hamiltonian system

Gagnon, L., Harnad, J., Winternitz, P. et Hurtubise, J., Abelian integrals and the reduction method for an integrable Hamiltonian system 26, 1605--1612 (1985), , J. Math. Phys.

#### Superposition laws for solutions of differential matrix Riccati equations arising in control theory

Sorine, Michel et Winternitz, Pavel, Superposition laws for solutions of differential matrix Riccati equations arising in control theory 30, 266--272 (1985), , IEEE Trans. Automat. Control

#### Separation of variables for the Hamilton-Jacobi equation on complex projective spaces

Boyer, C. P., Kalnins, E. G. et Winternitz, P., Separation of variables for the Hamilton-Jacobi equation on complex projective spaces 16, 93--109 (1985), , SIAM J. Math. Anal.

#### Nonlinear superposition principles: a new numerical method for solving matrix Riccati equations

Rand, D. W. et Winternitz, P., Nonlinear superposition principles: a new numerical method for solving matrix Riccati equations 33, 305--328 (1984), , Comput. Phys. Comm.

#### Classification of systems of nonlinear ordinary differential equations with superposition principles

Shnider, S. et Winternitz, P., Classification of systems of nonlinear ordinary differential equations with superposition principles 25, 3155--3165 (1984), , J. Math. Phys.

#### Comments on superposition rules for nonlinear coupled first-order differential equations

Winternitz, P., Comments on superposition rules for nonlinear coupled first-order differential equations 25, 2149--2150 (1984), , J. Math. Phys.

#### Symmetry reduction for nonlinear relativistically invariant equations

Grundland, A. M., Harnad, J. et Winternitz, P., Symmetry reduction for nonlinear relativistically invariant equations 25, 791--806 (1984), , J. Math. Phys.

#### Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras

Shnider, S. et Winternitz, P., Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras 8, 69--78 (1984), , Lett. Math. Phys.

#### Maximal abelian subalgebras of real and complex symplectic Lie algebras

Patera, J., Winternitz, P. et Zassenhaus, H., Maximal abelian subalgebras of real and complex symplectic Lie algebras 24, 1973--1985 (1983), , J. Math. Phys.

#### Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces

Boyer, C. P., Kalnins, E. G. et Winternitz, P., Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces 24, 2022--2034 (1983), , J. Math. Phys.

#### Normal forms of elements of classical real and complex Lie and Jordan algebras

Djokovi\'c, D. \v Z., Patera, J., Winternitz, P. et Zassenhaus, H., Normal forms of elements of classical real and complex Lie and Jordan algebras 24, 1363--1374 (1983), , J. Math. Phys.

#### Superposition principles for matrix Riccati equations

Harnad, J., Winternitz, P. et Anderson, R. L., Superposition principles for matrix Riccati equations 24, 1062--1072 (1983), , J. Math. Phys.

#### Solutions of the multidimensional sine-Gordon equation obtained by symmetry reduction

Grundland, A. M., Harnad, J. et Winternitz, P., Solutions of the multidimensional sine-Gordon equation obtained by symmetry reduction 4, 333--344 (1982), , Kinam Rev. F\'\i s.

#### Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation

Harnad, J. et Winternitz, P., Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation 23, 517--525 (1982), , J. Math. Phys.

#### Systems of ordinary differential equations with nonlinear superposition principles

Anderson, R. L., Harnad, J. et Winternitz, P., Systems of ordinary differential equations with nonlinear superposition principles 4, 164--182 (1981), , Phys. D

#### Group theoretical approach to superposition rules for systems of Riccati equations

Anderson, R. L., Harnad, J. et Winternitz, P., Group theoretical approach to superposition rules for systems of Riccati equations 5, 143--148 (1981), , Lett. Math. Phys.

#### Subgroups of Lie groups and separation of variables

Miller, Jr., W., Patera, J. et Winternitz, P., Subgroups of Lie groups and separation of variables 22, 251--260 (1981), , J. Math. Phys.

#### On the maximal abelian subgroups of the linear classical algebraic groups

Patera, J., Winternitz, P. et Zassenhaus, H., On the maximal abelian subgroups of the linear classical algebraic groups 2, 231--236 (1980), , C. R. Math. Rep. Acad. Sci. Canada

#### On the maximal abelian subgroups of the quadratic classical algebraic groups

Patera, J., Winternitz, P. et Zassenhaus, H., On the maximal abelian subgroups of the quadratic classical algebraic groups 2, 237--242 (1980), , C. R. Math. Rep. Acad. Sci. Canada

#### Discrete two-variable expansions of scattering amplitudes for particles with spin

Daumens, M. et Winternitz, P., Discrete two-variable expansions of scattering amplitudes for particles with spin 21, 1919--1927 (1980), , Phys. Rev. D (3)

#### Quadratic Hamiltonians in phase space and their eigenstates

Moshinsky, M. et Winternitz, P., Quadratic Hamiltonians in phase space and their eigenstates 21, 1667--1682 (1980), , J. Math. Phys.

#### Polynomial irreducible tensors for point groups

Patera, J., Sharp, R. T. et Winternitz, P., Polynomial irreducible tensors for point groups 19, 2362--2376 (1978), , J. Math. Phys.

#### Sous-algèbres de Lie de l'algèbre de Schrödinger

Burdet, G., Patera, J., Perrin, M. et Winternitz, P., Sous-algèbres de Lie de l'algèbre de Schrödinger 2, 81--108 (1978), , Ann. Sci. Math. Québec

#### Tensor fields invariant under subgroups of the conformal group of space-time

Beckers, J., Harnad, J., Perroud, M. et Winternitz, P., Tensor fields invariant under subgroups of the conformal group of space-time 19, 2126--2153 (1978), , J. Math. Phys.

#### The optical group and its subgroups

Burdet, G., Patera, J., Perrin, M. et Winternitz, P., The optical group and its subgroups 19, 1758--1780 (1978), , J. Math. Phys.

#### Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups

Patera, J., Sharp, R. T., Winternitz, P. et Zassenhaus, H., Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups 18, 2259--2288 (1977), , J. Mathematical Phys.

#### Subalgebras of real three- and four-dimensional Lie algebras

Patera, J. et Winternitz, P., Subalgebras of real three- and four-dimensional Lie algebras 18, 1449--1455 (1977), , J. Mathematical Phys.

#### Symmetry breaking interactions for the time dependent Schrödinger equation

Boyer, C. P., Sharp, R. T. et Winternitz, P., Symmetry breaking interactions for the time dependent Schrödinger equation 17, 1439--1451 (1976), , J. Mathematical Phys.

#### On bases for irreducible representations of $O(3)$ suitable for systems with an arbitrary finite symmetry group

Patera, J. et Winternitz, P., On bases for irreducible representations of $O(3)$ suitable for systems with an arbitrary finite symmetry group 65, 2725--2731 (1976), , J. Chem. Phys.

#### The group ${\rm O}(4)$, separation of variables and the hydrogen atom

Kalnins, E. G., Miller, Jr., W. et Winternitz, P., The group ${\rm O}(4)$, separation of variables and the hydrogen atom 30, 630--664 (1976), , SIAM J. Appl. Math.

#### Subgroups of the Poincaré group and their invariants

Patera, J., Sharp, R. T., Winternitz, P. et Zassenhaus, H., Subgroups of the Poincaré group and their invariants 17, 977--985 (1976), , J. Mathematical Phys.

#### Quantum numbers for particles in de Sitter space

Patera, J., Winternitz, P. et Zassenhaus, H., Quantum numbers for particles in de Sitter space 17, 717--728 (1976), , J. Mathematical Phys.

#### Subgroups of the similitude group of three-dimensional Minkowski space

Patera, J., Winternitz, P., Sharp, R. T. et Zassenhaus, H., Subgroups of the similitude group of three-dimensional Minkowski space 54, 950--961 (1976), , Canad. J. Phys.

#### Invariants of real low dimension Lie algebras

Patera, J., Sharp, R. T., Winternitz, P. et Zassenhaus, H., Invariants of real low dimension Lie algebras 17, 986--994 (1976), , J. Mathematical Phys.

#### Canonical transformation and accidental degeneracy. III. A unified approach to the problem

Moshinsky, M., Patera, J. et Winternitz, P., Canonical transformation and accidental degeneracy. III. A unified approach to the problem 16, 82--92 (1975), , J. Mathematical Phys.

#### Everything you always wanted to know about $SU(3)\supset {\rm O}(3)$

Moshinsky, M., Patera, J., Sharp, R. T. et Winternitz, P., Everything you always wanted to know about $SU(3)\supset {\rm O}(3)$ 95, 139--169 (1975), , Ann. Physics

#### Continuous subgroups of the fundamental groups of physics. II. The similitude group

Patera, J., Winternitz, P. et Zassenhaus, H., Continuous subgroups of the fundamental groups of physics. II. The similitude group 16, 1615--1624 (1975), , J. Math. Phys.

#### Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group

Patera, J., Winternitz, P. et Zassenhaus, H., Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group 16, 1597--1614 (1975), , J. Math. Phys.

#### Nagel-Moshinsky operators for discrete unitary representations of ${\rm U}(p, q)$

Patera, J., Winternitz, P. et Sharp, R. T., Nagel-Moshinsky operators for discrete unitary representations of ${\rm U}(p, q)$ 23, 81--98 (1974), , Rev. Mexicana F\'\i s.

#### Complete sets of commuting operators and $O(3)$ scalars in the enveloping algebra of $S{\rm U}(3)$

Judd, B. R., Miller, Jr., W., Patera, J. et Winternitz, P., Complete sets of commuting operators and $O(3)$ scalars in the enveloping algebra of $S{\rm U}(3)$ 15, 1787--1799 (1974), , J. Mathematical Phys.

#### The maximal solvable subgroups of ${\rm SO}(p,\,q)$ groups

Patera, J., Winternitz, P. et Zassenhaus, H., The maximal solvable subgroups of ${\rm SO}(p,\,q)$ groups 15, 1932--1938 (1974), , J. Mathematical Phys.

#### The maximal solvable subgroups of the ${\rm SU}(p,\,q)$ groups and all subgroups of ${\rm SU}(2,\,1)$

Patera, J., Winternitz, P. et Zassenhaus, H., The maximal solvable subgroups of the ${\rm SU}(p,\,q)$ groups and all subgroups of ${\rm SU}(2,\,1)$ 15, 1378--1393 (1974), , J. Mathematical Phys.

#### Potential scattering and Galilei-invariant expansions of scattering amplitudes

Kalnins, E. G., Patera, J., Sharp, R. T. et Winternitz, P., Potential scattering and Galilei-invariant expansions of scattering amplitudes 8, 3527--3538 (1973), , Phys. Rev. D (3)

#### Two-variable Galilei-group expansions of nonrelativistic scattering amplitudes

Kalnins, E. G., Patera, J., Sharp, R. T. et Winternitz, P., Two-variable Galilei-group expansions of nonrelativistic scattering amplitudes 8, 2552--2572 (1973), , Phys. Rev. D (3)

#### Invariant trilinear couplings involving both ${\rm SU}(2)$ and ${\rm SU}(1,\,1)$ states

Patera, J. et Winternitz, P., Invariant trilinear couplings involving both ${\rm SU}(2)$ and ${\rm SU}(1,\,1)$ states 14, 1977--1983 (1973), , J. Mathematical Phys.

#### One-parameter subgroups of unitary groups with indefinite metric and in particular of the conformal group

Belinfante, J. G. F. et Winternitz, P., One-parameter subgroups of unitary groups with indefinite metric and in particular of the conformal group 12, 1041--1054 (1971), , J. Mathematical Phys.

#### Crossing symmetric expansions of physical scattering amplitudes: the ${\rm O}(2,\,1)$ group and Lamé functions

Macfadyen, N. W. et Winternitz, P., Crossing symmetric expansions of physical scattering amplitudes: the ${\rm O}(2,\,1)$ group and Lamé functions 12, 281--293 (1971), , J. Mathematical Phys.

#### Recursion and symmetry relations for the Clebsch-Gordan coefficients of the homogeneous Lorentz group

Anderson, R. L., Raczka, R., Rashid, M. A. et Winternitz, P., Recursion and symmetry relations for the Clebsch-Gordan coefficients of the homogeneous Lorentz group 11, 1059--1068 (1970), , J. Mathematical Phys.

#### Clebsch-Gordan coefficients for the coupling of ${\rm SL}(2,\,C)$ principal-series representations

Anderson, R. L., Raczka, R., Rashid, M. A. et Winternitz, P., Clebsch-Gordan coefficients for the coupling of ${\rm SL}(2,\,C)$ principal-series representations 11, 1050--1058 (1970), , J. Mathematical Phys.

#### Representations of the Lorentz group: New integral relations between Legendre functions

Pajas, P. et Winternitz, P., Representations of the Lorentz group: New integral relations between Legendre functions 11, 1505--1510 (1970), , J. Mathematical Phys.

#### Two-dimensional expansions of relativistic amplitudes in the Mandelstam triangle and crossing symmetric reactions

Winternitz, P., Two-dimensional expansions of relativistic amplitudes in the Mandelstam triangle and crossing symmetric reactions 19, 1589--1601 (1969), , Czechoslovak J. Phys. Sect. B

#### Symmetry groups in classical and quantum mechanics

Winternitz, P., Smorodinski Ja. A., Uhlí\v r, M. et Fri\v s, I., Symmetry groups in classical and quantum mechanics 4, 444--450 (1967), , Soviet J. Nuclear Phys.

#### On relativistic angular momentum theory

Winternitz, P., Smorodinski, Ja. A. et Uhlí\v r, M., On relativistic angular momentum theory 1, 113--119 (1965), , Soviet J. Nuclear Phys.

#### Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group

Winternitz, P. et Fri\v s, I., Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group 1, 636--643 (1965), , Soviet J. Nuclear Phys.

#### On higher symmetries in quantum mechanics

Fri\v s, I., Mandrosov, V., Smorodinsky, Ja. A., Uhlí\v r, M. et Winternitz, P., On higher symmetries in quantum mechanics 16, 354--356 (1965), , Phys. Lett.

#### The optical theorem for scattering of particles with arbitrary spins

Winternitz, P., The optical theorem for scattering of particles with arbitrary spins 19, 1422--1424 (1964), , Soviet Physics JETP