  The domain ${\cal D}$ of all the coupling strengths compatible with the
reality of the energies is studied for a family of non-Hermitian $N$ by $N$
matrix Hamiltonians $H^{(N)}$ with tridiagonal and ${\cal PT}-$symmetric
structure. At all dimensions $N$, the coordinates are found of the extremal
points at which the boundary hypersurface $\partial {\cal D}$ touches the
circumscribed sphere (for odd $N=2M+1$) or ellipsoid (for even $N=2K$).
