  A toy-model quantum system is proposed. At a given integer $N$ it is defined
by the pair of $N$ by $N$ real matrices $(H,\Theta)$ of which the first item
$H$ specifies an elementary, diagonalizable non-Hermitian Hamiltonian $H \neq
H^\dagger$ with the real and explicit spectrum given by the zeros of the $N-$th
Chebyshev polynomial of the first kind. The second item $\Theta\neq I$ must be
(and is being) constructed as the related Hilbert-space metric which specifies
the (in general, non-unique) physical inner product and which renders our
toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation
$H^\dagger \Theta= \Theta\,H$. The elements of the (in principle, complete) set
of the eligible metrics are then constructed in closed band-matrix form. They
vary with our choice of the $N-$plet of optional parameters,
$\Theta=\Theta(\vec{\kappa})>0$ which must be (and are being) selected as lying
in the positivity domain of the metric, $\vec{\kappa} \in {\cal
D}^{(physical)}$.
