Conservation Laws in Field Dynamics or Why Boundary Motion is Exactly Integrable?

An infinite number of conserved quantities in the field dynamics $\phi_t = L U(\phi) + \rho$ for a linear Hermitian (or anti-Hermitian) operator $L$, an arbitrary function $U$ and a given source $\rho$ are presented. These integrals of motion are the multipole moments of the potential created by $\phi$ in the far-field. In the singular limit of a bistable scalar field $\phi = \phi_{\pm}$ (i.e. Ising limit) this theory describes a dissipative boundary motion (such as Stefan or Saffman-Taylor problem that is the continuous limit of the DLA-fractal growth) and can be exactly integrable. These conserved quantities are the polynomial conservation laws attributed to the integrability. The criterion for integrability is the uniqueness of the inverse potential problem's solution.