Realization of geodesic flow in the $k$-product monopoly

The present paper is devoted to the topological treatment of the variational model of $k$-product absolute monopoly. The class of models with the flow of optimal trajectories geodesically equivalent to the flow on plane is described. Stability of optimal trajectories and the possibility of existence of optimal synthesis are considered for periodical and non-periodical cases. A number of integrable cases are considered for $k=2$.