  In recent years, many exceptional orthogonal polynomials (EOP) were
introduced and used to construct new families of 1D exactly solvable quantum
potentials, some of which are shape invariant. In this paper, we construct from
Hermite and Laguerre EOP and their related quantum systems new 2D
superintegrable Hamiltonians with higher-order integrals of motion and the
polynomial algebras generated by their integrals of motion. We obtain the
finite-dimensional unitary representations of the polynomial algebras and the
corresponding energy spectrum. We also point out a new type of degeneracies of
the energy levels of these systems that is associated with holes in sequences
of EOP.
