Equigeodesics on Generalized Flag Manifolds with Two Isotropy Summands

In this paper we study homogeneous curves in generalized flag manifolds with two isotropy summands with the additional property that such curves are geodesics with respect to each invariant metric on the flag manifold. These curves are called equigeodesics. We give an algebraic characterization for such curves and we exhibit families of equigeodesics in several flag manifolds of classical and exceptional Lie groups.