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A natural generalization of a homogeneous geodesic on homogeneous Riemannianspaces $G/H$, which is called a two-step homogeneous geodesic, is a geodesic ofthe form $\gamma(t)=\pi(\exp(tx)\exp(ty))$, where $x,y$ belongs to the Liealgebra of $G$. In this paper, we extend this concept to homogeneous Finslerspaces. We give some sufficient conditions for $(\alpha,\beta)$-spaces andcubic spaces to admit a one-parameter family of invariant Finsler metrics to betwo-step Finsler geodesic orbit space. Also we give some examples of thesespaces.

Two-step homogeneous geodesics in some homogeneous Finsler manifolds