  The properties of a special configuration of a helium-like atomic system,
when both electrons are on the surface of a sphere of radius $r$, and angle
$\theta$ characterizes their positions on sphere, are investigated. Unlike the
previous studies, $r$ is considered as a quantum mechanical variable but not a
parameter. It is important that the "electrons-on-sphere" and the "collinear"
configuration are coincident in two points. For $\theta=0$ one obtains the
state of the electron-electron coalescence, whereas the angle $\theta=\pi$
characterizes the $\textbf{e-n-e}$ configuration when the electrons are located
at the ends of the diameter of sphere with the nucleus at its center. The
Pekeris-like method representing a fully three-body variational technique is
used for the expedient calculations. Some interesting features of the
expectation values representing the basic characteristic of the
"electrons-on-sphere" configuration are studied. The unusual properties of the
expectation values of the operators associated with the kinetic and potential
energy of the two-electron atom/ion possessing the "electrons-on-sphere"
configuration are found. Refined formulas for calculations of the two-electron
Fock expansion by the Green's function approach are presented. The analytic
wave functions of high accuracy describing the "electrons-on-sphere"
configuration are obtained. All results are illustrated in tables and figures.
