It is well known that equilibrium in a cosymmetric system in the general position is a member of a one-parameter family. In the present paper the Lyapunov-Schmidt method and the method of the central manifold are used to analyze bifurcations of such a family of equilibria as well as internal bifurcations: transitions of the type focus-node, node-saddle, and so on during motion along the family. A series of scenarios of branching of families of equilibria and the change in the structure of their arcs, consisting of equilibria of the same type, is described. Bifurcations of stable and unstable arcs, coalescence and decomposition of families of equilibria, bifurcation of the loss of smoothness by the family of equilibria, and branching of a small equilibrium cycle from a corner point of the family of equilibria are investigated in detail. The variability of the spectrum along a family gives rise to a variety of new phenomena that are not encountered in the classical case of an isolated equilibrium or in bifurcations of families of equilibria of a system with symmetry. They include protraction with respect to the branching parameter of the family of equilibria, Lyapunov instability of a family of equilibria with the attraction properties being preserved, and the appearance and disappearance of new stable and unstable arcs on the family of equilibria. (c) 2000 American Institute of Physics.