The Novikov – Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in C P 2

We associate a periodic two-dimensional Schrödinger operator to every Lagrangian torus in CP 2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov–Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP 2 preserving the spectral curve. We also show that the highest flows on the space of smooth periodic solutions of the Tzizéica equation are given by the Novikov–Veselov hierarchy.