  An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq
[ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming that
each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it
ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an
explicit specification of all of these spaces (i.e., in essence, of all of the
eligible {\it ad hoc} inner products and metric operators $\Theta$) is
addressed. The problem is shown exactly solvable and, for every size
$N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda)$,
the complete $N-$parametric set of metrics
$\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined
by closed formula.
