Supercritical spatially homogeneous branching in R-n admits no equilibria

At the beginning of investigations in spatially homogeneous branching proce sses in Euclidean space (LIEMANT [1]) it Seemed to be obvious that the exis tence of equilibria implies criticality of branching. This prejudice was di sproved by the example [2] of a subcritical homogeneous branching equilibri um in dimension one. We prove that supercritical homogeneous branching proc esses in Euclidean space and, more general, in a broad class of topological groups have no (non - void, homogeneous) equilibria.