  A discrete $N-$point Runge-Kutta version $H^{(N)}({\lambda})$ of one of the
simplest non-Hermitian square-well Hamiltonians with real spectrum is studied.
A complete set of its possible hermitizations (i.e., of the eligible metrics
$\Theta^{(N)}({\lambda})$ defining its non-equivalent physical Hilbert spaces
of states) is constructed, in closed form, for any coupling ${\lambda}\in
(-1,1)$ and any matrix dimension $N$.
