Let X and Y be real Banach spaces, let f: Rp × X ? Y be a Cl mapping, l = 2, and let L(?) = Df(?, 0), for ? ? Rp. Suppose also that f(?, 0) = 0, for all ?, and at some point ?0, dim N(L(?0)) = n > 0, codim R(L(?0)) = m > 0. We examine the structure of the zero set of f in a neighbourhood of (?0, 0), and give conditions under which this zero set is diffeomorphic to the zero set of the lower order terms in the Taylor expansion of f near (?0, 0). These results tell us about the structure of the set of non-trivial solutions bifurcating from zero when there are several parameters and the dimension of the null space of the linearization is greater than 1. Finally, we consider the genericity of the conditions used. © 1995 Academic Press. All rights reserved.