  The angular coefficients $\psi_{k,p}(\alpha,\theta)$ of the Fock expansion
characterizing the S-state wave function of the two-electron atomic system, are
calculated in hyperspherical angular coordinates $\alpha$ and $\theta$. To
solve the problem the Fock recurrence relations separated into the independent
individual equations associated with definite power $j$ of the nucleus charge
$Z$, are applied. The "pure" $j$-components of the angular Fock coefficients,
orthogonal to of the hyperspherical harmonics $Y_{kl}$, are found for even
values of $k$. To this end, the specific coupling equation is proposed and
applied. Effective techniques for solving the individual equations with
simplest nonseparable and separable right-hand sides are proposed. Some
mistakes/misprints made earlier in representations of $\psi_{2,0}$, were noted
and corrected. All $j$-components of $\psi_{4,1}$ and the majority of
components and subcomponents of $\psi_{3,0}$ are calculated and presented for
the first time. All calculations were carried out with the help of the Wolfram
\emph{Mathematica}.
