  Coupled cell systems are dynamical systems associated to a network and
synchrony subspaces, given by balanced colorings of the network, are invariant
subspaces for every coupled cell systems associated to that network. Golubitsky
and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an
analogue of the Equivariant Branching Lemma in the context of regular networks.
We generalize this result proving the generic existence of steady-state
bifurcation branches for regular networks with maximal synchrony. We also give
necessary and sufficient conditions for the existence of steady-state
bifurcation branches with some submaximal synchrony. Those conditions only
depend on the network structure, but the lattice structure of the balanced
colorings is not sufficient to decide which synchrony subspaces support a
steady-state bifurcation branch.
