On the multiplicity of isometry-invariant geodesics on product manifolds

We prove that on any closed Riemannian manifold $(M_1\times M_2,g)$, with $\text{rank}{\text{Hom}}_1(M_1)\mathrm{\ne }0$ and $\dim (M_2)\ge 2$, every isometry homotopic to the identity admits infinitely many isometry-invariant geodesics.