  Dynamics of planar domains with moving boundaries driven by the gradient of a
scalar field that satisfies an elliptic PDE is studied. We consider the
question: For which kind of PDEs the domains are algebraic, provided the field
has singularities at a fixed point inside the domain? The construction reveals
a direct connection with the theory of the Calogero-Moser systems related to
finite reflection groups and their integrable deformations.
