  We consider the model of strongly-correlated system of electrons described by
an extended Falicov-Kimball Hamiltonian where the stability of some axial and
diagonal striped phases was proved. Introducing a next-nearest-neighbor
hopping, small enough not to destroy the striped structure, we examine
rigorously how the presence of the next-nearest-neighbor hopping anisotropy
reduces the $\pi/2$-rotation degeneracy of the diagonal-striped phase. The
effect appears to be similar to that in the case of anisotropy of the
nearest-neighbor hopping: the stripes are oriented in the direction of the
weaker next-nearest-neighbor hopping.
