N-fold Supersymmetry and Quasi-solvability Associated with X₂-Laguerre Polynomials

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.