  We consider a Coulomb system of one electron and five or six infinitely
massive centers of charge $Z$: $(5Z,e)$ and $(6Z,e)$. Critical charges and the
possible optimal geometrical configurations are found. It is shown that the
domain of stability for $(5Z,e)$ is $0 < Z \leq Z_{cr}^{(5Z,e)}=0.350$ with the
optimal geometrical configuration given by a dipyramid (equilateral triangle
base) circumscribed in a prolate spheroid. For $(6Z,e)$ the stability is $0 < Z
\leq Z_{cr}^{(6Z,e)}=0.335$ with the optimal geometrical configuration given by
an octahedron (square base), circumscribed in an oblate spheroid. For both
systems we obtain an
  indication that total energy at $Z=Z_{cr}$ has a square-root branch point
singularity with exponent $3/2$.
