  We introduce a new family of quasi-exactly solvable generalized isotonic
oscillators which are based on the pseudo-Hermite exceptional orthogonal
polynomials. We obtain exact closed-form expressions for the energies and
wavefunctions as well as the allowed potential parameters for the first two
members of the family using the Bethe ansatz method. Numerical calculations of
the energies reveal that member potentials have multiple quasi-exactly solvable
eigenstates and the number of states for higher members are parameter
dependent.
