Canonical affinor structures of classical type on regular $\Phi$-spaces

For arbitrary regular $\Phi$-spaces all canonical affinor structures of classical type, that is, the almost product, almost complex, and, more generally, $f$-structures ($f^3+f=0$), are described. Criteria for existence are indicated and computation algorithms for such structures are presented. In particular, for homogeneous $\Phi$-spaces of arbitrary finite order, precise computational formulae are indicated, which were earlier for $n=3$ and (partially) for $n=5$. All the above-mentioned geometric result are obtained using the complete solution of a general algebraic problem about the roots of the equations $x^2=\pm1$ and $x^3+x=0$ in the quotient ring of polynomials and in the corresponding operator ring.