Steady states of many lumped parameter systems are described by equations that cannot be reduced to a single equation. Existing works on the bifurcation diagrams and steady-state multiplicity either assume that the equations are reducible to one equation or use some mathematical techniques (e.g. Lyapounov reduction) to reduce them. In an earlier communication (Inamdar and Karimi (Chem. Eng. Sci. 56(12) (2001) 3915), we introduced the use of a new and powerful method, called reductive perturbation method (RPM), which deals with multiple equations as they are, without using any reduction. In this work, we apply RPM to fully study the branching and stability of steady states near a singularity in an n-equation system. Analogous to those known for the one-equation system, we identify several types of singularities and branching patterns for the n-equation system, including some that have not appeared so far in the chemical engineering literature. We present explicit, analytical conditions and expressions for the occurrence of these singularities, the stationary solution branches and their stability. Our work easily generalizes and complements many of the branching and multiplicity results available for the one-equation system and is the first to give explicit, analytical results on branch stability. As illustrated using a literature example, it also facilitates computation and construction of solution diagrams.