  We obtain necessary and sufficient conditions for the integrability in
quadratures of geodesic flows on homogeneous spaces $M$ with invariant and
central metrics. The proposed integration algorithm consists in using a special
canonical transformation in the space $T^*M$ based on constructing the
canonical coordinates on the orbits of the coadjoint representation and on the
simplectic sheets of the Poisson algebra of invariant functions. This algorithm
is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie
group.
