Non-polynomial integrals of multidimensional geodesic flows and Lie algebras

In this paper, we construct explicit local examples of multidimensional Riemannian metrics whose geodesic flows have non-polynomial first integrals and are completely integrable. We rely on a construction described in a recent paper by A.V. Galajinsky which allows one to construct such examples via the Casimir invariants of finite-dimensional Lie algebras.