  To capture a multidimensional consistency feature of integrable systems in
terms of the geometry, we give a condition called \emph{geodesic compatibility}
that implies the existence of integrals in involution of the geodesic flow. The
geodesic compatibility condition is constructed from a concrete example namely
the integrable Calogero's Goldfish system through the Poisson structure and the
variational principle. The geometrical view of the geodesic compatibility gives
a compatible parallel transport between two different Hamiltonian vector
fields.
