  We investigate the breakdown of normal hyperbolicity of a manifold of
equilibria of a flow. In contrast to classical bifurcation theory we assume the
absence of any flow-invariant foliation at the singularity transverse to the
manifold of equilibria. We call this setting bifurcation without parameters. In
the present paper we provide a description of general systems with a manifold
of equilibria of codimension one as a first step towards a classification of
bifurcations without parameters. This is done by relating the problem to
singularity theory of maps.
