  We study non-linear integrable partial differential equations naturally
arising as bi-Hamiltonian Euler equations related to the looped cotangent
Virasoro algebra. This infinite-dimensional Lie algebra (constructed in
\cite{OR}) is a generalization of the classical Virasoro algebra to the case of
two space variables. Two main examples of integrable equations we obtain are
quite well known. We show that the relation between these two equations is
similar to that between the Korteweg-de Vries and Camassa-Holm equations.
