Riemannian g.o. metrics in certain M-spaces

Let G/K be a generalized flag manifold with K=C(S)=S×K1, where S is a torus in a compact simple Lie group G and K1 is the semisimple part of K. Then the associated M-space is the homogeneous space G/K1. These spaces were introduced and studied by H.C. Wang in 1954. We mainly investigate homogeneous geodesics in M-spaces which correspond to generalized flag manifolds with two isotropy summands and we also give some results for M-spaces corresponding to flag manifolds with at least three isotropy summands, thus extending some previous results. Finally, we prove that there are several M-spaces which admit nonnaturally reductive g.o. metrics.