  A spectral sequence calculating the homology groups of some spaces of maps
equivariant under compact group actions is described. For the main example, we
calculate the rational homology groups of spaces of even and odd maps $S^m \to
S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of
non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb
R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of
${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a
technical tool, we calculate the homology groups of configuration spaces of
projective and lens spaces with coefficients in certain local systems.
