Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.