  We consider Hamiltonians, which are even polynomials of the forth order with
the respect to Bose operators. We find subspaces, preserved by the action of
Hamiltonian These subspaces, being finite-dimensional, include, nonetheless,
states with an \QTR{it}{infinite} number of quasi-particles, corresponding to
the original Bose operators. The basis functions look rather simple in the
coherent state representation and are expressed in terms of the degenerate
hypergeometric function with respect to the complex variable labeling the
representation. In some particular degenerate cases they turn (up to the power
factor) into the trigonometric or hyperbolic functions, Bessel functions or
combinations of the exponent and Hermit polynomials. We find explicitly the
relationship between coefficients at different powers of Bose operators that
ensure quasi-exact solvability of Hamiltonian.
