Canonical distributions on Riemannian homogeneous kk-symmetric spaces

It is known that distributions generated by almost product structures are applicable, in particular, to some problems in the theory of Monge–Ampère equations. In this paper, we characterize canonical distributions defined by canonical almost product structures on Riemannian homogeneous kk-symmetric spaces in the sense of types AF (anti-foliation), F (foliation), TGF (totally geodesic foliation). Algebraic criteria for all these types on kk-symmetric spaces of orders k=4,5,6k=4,5,6 were obtained. Note that canonical distributions on homogeneous kk-symmetric spaces are closely related to special canonical almost complex structures and ff-structures, which were recently applied by I. Khemar to studying elliptic integrable systems.