  We demonstrate a novel approach that allows the determination of very general
classes of exactly solvable Hamiltonians via Bethe ansatz methods. This
approach combines aspects of both the co-ordinate Bethe ansatz and algebraic
Bethe ansatz. The eigenfunctions are formulated as factorisable operators
acting on a suitable reference state. Yet, we require no prior knowledge of
transfer matrices or conserved operators. By taking a variational form for the
Hamiltonian and eigenstates we obtain general exact solvability conditions. The
procedure is conducted in the framework of Hamiltonians describing the
crossover between the low-temperature phenomena of superconductivity, in the
Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einstein condensation (BEC).
