Many applications give rise to separable parameterized equations of the form A(y,µ)z + b(y, µ) = 0, where y ∈ Rn, z ∈ RN and the parameter µ ∈ R; here A(y,µ) is an (N + n) × N matrix and b(y, µ) ∈ RN +n. Under the assumption that A(y, µ) has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form f (y, µ) = 0. In this paper we extend that method to the case that A(y, µ) has rank deficiency one at the bifurcation point. At such a point the solution curve (y, µ, z) branches into infinitely many additional solutions,which form a straight line. A numerical method for reducing the problem to a smaller space and locating such a bifurcation point is given. Applications to equilibrium solutions of nonlinear ordinary equations and solutions of discretized partial differential equations are provided.