Let $u:[0,T] \to D$ be a $C^2$ Hilbert-space-valued function satisfying an “abstract” wave equation of the form $Pu_{tt} = - Au + \mathcal{F}(u(t))$, where P and A are symmetric linear operators defined on D (a dense linear subspace of a Hilbert space H) and where $\mathcal{F}:D \to H$ is a gradient operator with potential $\mathcal{G}:D \to R$. Let $\mathcal{G}$ be almost homogeneous of degree $4\alpha + 2$ for some $\alpha > 0((4\alpha + 2)\mathcal{G}(x) \leqq (x,\mathcal{F}(x)))$ for all $x \in D$ and let $u(0) = u_0 $, $u_t (0) = v_0 $. We extend the results of [1] to the case $E_\Sigma (0) \equiv \frac{1}{2}(u_0 ,Pu_0 ) + \frac{1}{2}(v_0 ,Pv_0 ) - \mathcal{G}(u_0 ) = 0$ and $(u_0 ,Pv_0 ) \leqq 0$. We prove the following.If $E_\Sigma (0) = 0$, $(u_0, Pv_0) = 0$ and $u_0 \ne 0$, then u cannot exist on $[0,\infty )$ in the sense that there exists $T $, $0 < T < \infty $, such that $\lim _{t \to T - } (u(t),Pu(t)) = + \infty $. If $E_\Sigma (0) = 0$ and $(u_0 ,Pv_0 ) < 0$, then either u exists on $[0,\in...