  We construct a new family of quasi-solvable and N-fold supersymmetric quantum
systems where each Hamiltonian preserves an exceptional polynomial subspace of
codimension 2. We show that the family includes as a particular case the
recently reported rational radial oscillator potential whose eigenfunctions are
expressed in terms of the X_2-Laguerre polynomials of the second kind. In
addition, we find that the two kinds of the X_2-Laguerre polynomials are
ingeniously connected with each other by the N-fold supercharge.
