||To study symmetric properties of solutions to equivariant variational problems, Kazimierz Geba introduced the so-called $G$-equivariant
gradient degree taking its values in the Euler ring $U(G)$. In my talk I will indicate several techniques to evaluate the multiplication structure of the Euler ring $U(\Gamma\times S^1)$, where $\Gamma$ is a compact Lie group. In addition, some methods for the computation of the $\Gamma\times
S^1$-equivariant degree, based on its connections with other equivariant degrees. Finally, the obtained results will be applied to an asymptotically linear variational autonomous Newtonian system with symmetries for which we will prove the existence of multiple periodic solutions with various symmetry properties.