The author considers the relationship between the blow-out bifurcation and another bifurcation which creates a pair of stable invariant curves and can properly be called a generalized pitchfork bifurcation. The goal of this paper is to show that the two types of bifurcation (generalized) described above are really part of the same generalized codimension two bifurcation. The author constructs an example for which all the bifurcation curves can be calculated analytically, making it possible to provide much stronger numerical confirmation that the second type of bifurcation occurs rather than by a process of fractalization.