This fold catastrophe appears also in general relativity in which $A(\varphi)=1$ identically. In general relativity, because of the absence of a gravitational scalar field, the control space is one-dimensional. For $\beta=-4$, the maximum ADM energy is greater than the general relativistic one for $\varphi_0\neq 0$. This is because, due to the presence of the scalar field, the effective gravitational constant becomes smaller and thereby gravity becomes weaker than in general relativity.

$\beta\protect\alt -4.35$ case
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This case is more interesting than the above case. We present the results of the case $\beta=-6$. Figure 3 shows $\tilde{\rho}_c=\mbox{const}$ curves in the $(\varphi_0,N)$ plane. This figure is very different from Fig. 1. On the envelope of the family of curves, $e^{\prime}dcbab^{\prime}c^{\prime}de$, the sequence of equilibrium solutions changes its stability. Although there are other envelopes in region (B), they have nothing to do with the change of the number of stable equilibrium solutions. Figure 4 shows curves $(\varphi_0,\omega)$ with $N$ constant. From criteria (i) and (ii), the number of stable equilibrium solutions is as follows: In region (A), only one stable equilibrium solution exists. In region (B), two distinct stable equilibrium solutions exist. Surprisingly, these stable equilibrium solutions are different even for $\varphi_0=0$ from 