The standard object for vector fields with a nontrivial cosymmetry is a continuous one-parameter family of equilibria. Characteristically, the stability spectrum of equilibrium varies along such a family, though the spectrum always contains a zero point. Consequently, in the general position a family consists of stable and unstable arcs separated by boundary equilibria, which are neutrally stable in the linear approximation. In the present paper the central manifold method and the Lyapunov-Schmidt method are used to investigate the branching bifurcation of invariant two-dimensional tori in cosymmetric systems off a boundary equilibrium whose spectrum contains, besides the requisite point 0, two pairs of purely imaginary eigenvalues. A number of new effects, as compared with the classic case of an isolated equilibrium, are found: the bifurcation studied has codimension 1 (2 for an isolated equilibrium); it is accompanied by a branching bifurcation of a normal limit cycle; and, a stable arc can be created on an unstable arc. (c) 2001 American Institute of Physics.