Polypolar coordination and symmetries

The polypolar system of coordinates is formed by a family of a parametrized on a radius isofocal of $kf$-lemniscates. As well as the classical polar system of coordinates, it characterizes a point of a plane by a polypolar radius $\rho$ and polypolar angle $\varphi$. For anyone connectedness a family isometric of curve $\rho=const$ — lemniscates and family gradient of curves $\varphi=const$ — are mutually orthogonal conjugate coordinate families. The singularities of polypolar coordination, its symmetry, and also curvilinear symmetries on multifocal lemniscates are considered.