A classical magnetic model with compact su(2) and non-compact su(1,1) spin phase space admitting the Hirota bilinear form is presented in an arbitrary number of space dimensions. The essential point of the construction is the presence of a velocity field with non-trivial vorticity tensor related to a topological current density. The model modifies in particular a variety of familiar topological equations, such as the Heisenberg ferromagnets and the (2+1)-dimensional O(3) sigma -model. Using the bilinear representation, several special cases and exact solutions of physical interest (spin waves, domain walls and vortices) are considered.