Before revision:  We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with boundaries , while for a given value of the parameters, it is an embedded smooth and connected curve.
After revision:  We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners , while for a given value of the parameters, it is an embedded smooth and connected curve.