  We consider one variant of stable assignment problems in a bipartite graph
endowed with nonnegative capacities on the edges and quotas on the vertices. It
can be viewed as a generalization of the stable allocation problem introduced
by Ba\H{\i}ou and Balinsky, which arises when strong linear orders of
preferences on the vertices in the latter are replaced by weak ones. At the
same time, our stability problem can be stated in the framework of a theory by
Alkan and Gale on stable schedule matchings generated by choice functions of a
wide scope. In our case, the choice functions are of a special, so-called
mixed, type.
  The main content of this paper is devoted to a study of rotations in our
mixed model, functions on the edges determining ``elementary'' transformations
between close stable assignments. These look more sophisticated compared with
rotations in the stable allocation problem (which are generated by simple
cycles). We efficiently construct a poset of rotations and show that the stable
assignments are in bijection with the so-called closed functions for this
poset; this gives rise to a ``compact'' affine representation for the lattice
of stable assignments and leads to an efficient method to find a stable
assignment of minimum cost.
