Patterns of Non-Stationary Solutions to Symmetric Systems of Second-Order Differential Equations

We establish the existence of non-stationary solutions to a symmetric system of second-order autonomous differential equations. Our technique is based on the equivariant degree theory and involves a novel characterization of orbit types of maximal kind in the Burnside Ring product of a finite number of basic degrees for the group $O(2)\times \mathrm{\Gamma }\times {\mathbb{Z}}_2$.