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We consider a Coulomb system of one electron and five or six infinitelymassive centers of charge $Z$: $(5Z,e)$ and $(6Z,e)$. Critical charges and thepossible optimal geometrical configurations are found. It is shown that thedomain of stability for $(5Z,e)$ is $0 < Z \leq Z_{cr}^{(5Z,e)}=0.350$ with theoptimal geometrical configuration given by a dipyramid (equilateral trianglebase) 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 byan octahedron (square base), circumscribed in an oblate spheroid. For bothsystems we obtain an  indication that total energy at $Z=Z_{cr}$ has a square-root branch pointsingularity with exponent $3/2$.

Critical charge of a system with one electron and five or six charged centers