In this short review paper the detailed analysis of six two-dimensional quantum superintegrable systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, G_2 rational, or I_6) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index k. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure g^(k) with various k, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra g^(k) for a certain k. In all presented cases the algebra of integrals is a 4-generated (H, I_1, I_2, I_12≡[I_1, I_2]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.

This article is the third in a series, the aim of which is to use Lie group theory to obtain exact analytic solutions of delay ordinary differential systems (DODSs). Such a system consists of two equations involving one independent variable x and one dependent variable y . As opposed to ordinary differential equations (ODEs) the variable x figures in more than one point (we consider the case of two points, x and x − ). The dependent variable y and its derivatives figure in both x and x − . Two previous articles were devoted to first -order DODSs, here we concentrate on a large class of second -order ones. We show that within this class the symmetry algebra can be of dimension n with 0 ⩽ n ⩽ 6 for nonlinear DODSs and must be infinite-dimensional for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2,…, 5. Among the 6 ODEs identifying the Painlevé transcendents only P III , P V and P VI have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

We investigate a quantum non-relativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. Assuming that the Hamiltonian is rotationally invariant and parity conserving we identify all such systems which allow additional (pseudo)tensor integrals of motion that are second order matrix polynomials in the momenta. Previously we found all the (pseudo)scalar and (axial)vector integrals of motion. No non-obvious tensor integrals exist. However, nontrivial pseudo-tensor integrals do exist. Together with our earlier results we give a complete list of such superintegrable Hamiltonian systems allowing second-order integrals of motion.

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space E_2 allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian H and the second order integral of motion X, responsible for the separation of variables, they allow a third integral that is a polynomial of order N (N≥3) in the components p_1, p_2 of the linear momentum. We focus on doubly exotic potentials, i.e. potentials V(x, y) = V_1(x) + V_2(y) where neither V_1(x) nor V_2(y) satisfy any linear ordinary differential equation. We present two new infinite families of superintegrable systems in E_2 with integrals of order N for which V_1(x) and V_2(y) are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for 3≤ N ≤ 10. We conjecture that this will hold for any doubly exotic potential and for all N, and that moreover the potentials will always actually have the Painlevé property.

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order N > 2. Two types of such superintegrable potentials exist. The first type consists of “standard potentials” that satisfy linear differential equations. The second type consists of “exotic potentials” that satisfy nonlinear equations. For N = 3, 4, and 5 these equations have the Painlevé property. We conjecture that this is true for all N ≥ 3. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any N) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

Lie point symmetries of delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) are considered. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Moreover, the symmetry algebras of these linear or linearizable DODSs of order N contain a subalgebra of dimension dim L = 2N realized by linearly connected vector fields. Genuinely nonlinear DODSs of order N have symmetry algebras of dimension n, 0 ≤ n ≤ 2N + 2. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction. In particular we present invariant solutions of a DODS originating in a study of traffic flow.

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates.In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p.We assume that their Poisson commutator {H, K} vanishes, is a constant, a constant times H, or a constant times K.In the quantum case H and K are operators and their Lie commutator has one of the above properties.We use two copies of such (H, K) pairs to generate two-dimensional superintegrable systems in the Euclidean space E 2 , allowing the separation of variables in Cartesian coordinates.All known separable superintegrable systems in E 2 can be obtained in this manner and we obtain new ones for N = 4.

A recent article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs, which have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional subalgebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone, and we present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

Systems consisting of delay ordinary differential equations and delay relations are considered. If such a system is invariant with respect to a Lie group transformation, there is a possibility to look for particular solutions, which are invariant with respect to this transformation. Thus, we can find particular solutions.

A group classification of first-order delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs), which consists of linear DODEs and solution-independent delay relations, have infinite-dimensional symmetry algebras—as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension n, . It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E-2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr ö dinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. Their angular dependent part S ( θ ) satisfies a nonlinear ODE that has the Painlev é property and its solutions can be expressed in terms of the Painlev é transcendent P 6 . We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form . Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibility of finding superintegrable systems among the integrable ones (i.e. systems with 1 or 2 additional independent integrals).

We consider a two dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) nor V_2(y) satisfy a linear ODE) allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painleve property and integrate them in terms of known Painleve transcendents or elliptic functions.