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An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq[ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming thateach of these matrices represents a selfadjoint quantum Hamiltonian in an {\itad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of anexplicit specification of all of these spaces (i.e., in essence, of all of theeligible {\it ad hoc} inner products and metric operators $\Theta$) isaddressed. 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 definedby closed formula.

The complete menu of eligible metrics for a family of toy Hamiltonians $H \neq H^\dagger$ with real spectra