  We consider a homotopic evolution in the space of smooth shapes starting from
the unit circle. Based on the Loewner-Kufarev equation we give a Hamiltonian
formulation of this evolution and provide conservation laws. The symmetries of
the evolution are given by the Virasoro algebra. The `positive' Virasoro
generators span the holomorphic part of the complexified vector bundle over the
space of conformal embeddings of the unit disk into the complex plane and
smooth on the boundary. In the covariant formulation they are conserved along
the Hamiltonian flow. The `negative' Virasoro generators can be recovered by an
iterative method making use of the canonical Poisson structure. We study an
embedding of the Loewner-Kufarev trajectories into the Segal-Wilson
Grassmannian, construct the tau-function, the Baker-Akhiezer function, and
finally, give a class of solutions to the KP equation.
