  Exceptional orthogonal Hermite and Laguerre polynomials have been linked to
the k-step extension of harmonic and singular oscillators. The exceptional
polynomials allow the existence of different supercharges from the Darboux-Crum
and Krein-Adler constructions of supersymmetric quantum mechanics. They also
allow the existence of different types of ladder relations and their associated
recurrence relations. The existence of such relations is a unique property of
these polynomials. Those relations have been used to construct 2D models which
are superintegrable, and display an interesting spectrum, degeneracies and
finite-dimensional unitary representations. In previous works, only the
physical or polynomial part of the spectrum is discussed. It is known that the
general solutions are associated with other types of recurrence/ladder
relations. We plan to discuss in detail the case of the exceptional Hermite
polynomials $X_2^{(1)}$ and to explicitly present new chains obtained by acting
with different types of ladder operators. We will exploit a recent result by
one of the authors [32], where the general analytic solution was constructed
and connected with the non-degenerate confluent Heun equation. The analogue
Rodrigues formulas for the general solution are constructed. The set of finite
states from which the other states can be obtained algebraically is not unique
but the vanishing arrow and diagonal arrow from the diagram of the 2-chain
representations can be used to obtain minimal sets. These Rodrigues formulas
are then exploited, not only to construct the states, polynomial and
non-polynomial, in a purely algebraic way, but also to obtain coefficients from
the action of ladder operators also in an algebraic manner based on further
commutation relations between monomials of the generators.
