Title: Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Abstract: I will review results on classiﬁcation of quantum superintegrable systems on two-dimensional Euclidean space allowing separation of variables in Cartesian coordinates and possessing an extra integral of third or fourth order. The exotic quantum potential satisfy a nonlinear ODE and have been shown to exhibit the Painlevé property. I will also present diﬀerent constructions of higher order superintegrable Hamiltonians involving Painlev´e transcendents using four types of building blocks which consist of 1D Hamiltonians allowing operators of the type Abelian, Heisenberg, Conformal or Ladder. Their integrals generate ﬁnitely generated polynomial algebras and representations can be exploited to calculate the energy spectrum. I will point out that for certain cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. I will describe how other sets of integrals can be build and used to provide a complete solution.