Chemical reaction networks describing mass-action kinetics, with graph vertexes for the complexes and graph edges for the reactions, have been further developed to generalized reaction networks describing more general kinetics (such as power-law kinetics), so-called generalized mass-action systems, that mirror the stoichiometry by stoichiometric complexes and the kinetics by kinetic-order complexes, resp. The focus of study is on vertex-balanced steady states, i.e., steady states with net flux zero across every vertex, in particular on existence and uniqueness of these in every (invariant) stoichiometric subspace, for all rate constants. In the context of Birch’s theorem from statistics, if stoichiometric subspace and kinetic-order subspace have the same dimension, a condition (involving sign vectors) is found that guarantees existence and uniqueness of a vertex-balanced steady state in every stoichiometric subspace for any rate constants, provided that such a steady state exists for some rate constant. Then, moreover, the deficiency with respect to the kinetic-order complexes vanishes if and only if such a steady state exists for all rate constants. Finally, if the deficiency with respect to the stoichiometric complexes is zero, there is exactly one positive steady state, which is vertex-balanced.