Veselov-Novikov hierarchy of equations, and integrable deformations of minimal Lagrangian tori in $\mathbb CP^2$

We associate a periodic two-dimensional Schrödinger operator to every Lagrangian torus in $\mathbb CP^2$ and define the spectral curve of a torus as the Floquet spectrum on this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that the Novikov–Veselov hierarchy of equations induces integrable deformations of a minimal Lagrangian torus in $\mathbb CP^2$ preserving the spectral curve.