Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1

We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev--Petviashvili (of type B) equation.