  In the present paper we study locally semiflat (we also call them
semiintegrable) almost Grassmann structures. We establish necessary and
sufficient conditions for an almost Grassmann structure to be alpha- or
beta-semiintegrable. These conditions are expressed in terms of the fundamental
tensors of almost Grassmann structures. Since we are not able to prove the
existence of locally semiflat almost Grassmann structures in the general case,
we give many examples of alpha- and beta-semiintegrable structures, mostly
four-dimensional. For all examples we find systems of differential equations of
the families of integral submanifolds V_alpha and V_beta of the distributions
Delta_alpha and Delta_beta of plane elements associated with an almost
Grassmann structure. For some examples we were able to integrate these systems
and find closed form equations of submanifolds V_alpha and V_beta.
