  We introduce a general Hamiltonian describing coherent superpositions of
Cooper pairs and condensed molecular bosons. For particular choices of the
coupling parameters, the model is integrable. One integrable manifold, as well
as the Bethe ansatz solution, was found by Dukelsky et al., Phys. Rev. Lett. 93
(2004) 050403. Here we show that there is a second integrable manifold,
established using the boundary Quantum Inverse Scattering Method. In this
manner we obtain the exact solution by means of the algebraic Bethe ansatz. In
the case where the Cooper pair energies are degenerate we examine the
relationship between the spectrum of these integrable Hamiltonians and the
quasi-exactly solvable spectrum of particular Schrodinger operators. For the
solution we derive here the potential of the Schrodinger operator is given in
terms of hyperbolic functions. For the solution derived by Dukelsky et al.,
loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric
boundary conditions. This latter case provides a novel example of an integrable
Hermitian Hamiltonian acting on a Fock space whose states map in to a Hilbert
space of PT-symmetric wavefunctions defined on a contour in the complex plane.
