  Four new families of two-dimensional quantum superintegrable systems are
constructed from k-step extension of the harmonic oscillator and the radial
oscillator. Their wavefunctions are related with Hermite and Laguerre
exceptional orthogonal polynomials (EOP) of type III. We show that ladder
operators obtained from alternative construction based on combinations of
supercharges in the Krein-Adler and Darboux Crum ( or state deleting and
creating ) approaches can be used to generate a set of integrals of motion and
a corresponding polynomial algebra that provides an algebraic derivation of the
full spectrum and total number of degeneracies. Such derivation is based on
finite dimensional unitary representations (unirreps) and doesn't work for
integrals build from standard ladder operators in supersymmetric quantum
mechanics (SUSYQM) as they contain singlets isolated from excited states. In
this paper, we also rely on a novel approach to obtain the finite dimensional
unirreps based on the action of the integrals of motion on the wavefunctions
given in terms of these EOP. We compare the results with those obtained from
the Daskaloyannis approach and the realizations in terms of deformed oscillator
algebras for one of the new families in the case of 1-step extension. This
communication is a review of recent works.
