  In a pre-selected Hilbert space of quantum states the unitarity of the
evolution is usually guaranteed via a pre-selection of the generator (i.e., of
the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use
of both of these pre-selections is overrestrictive. One should be allowed to
make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization
of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity
scenario with nontrivial metric, the current concept of solvability (meaning,
most often, the feasibility of a non-numerical diagonalization of Hamiltonian)
requires a generalization allowing for a non-numerical form of metric. A few
illustrative solvable quantum models of this type are presented.
