  In this paper, we establish some results about the singular points of certain
non-monotone potential operators. Here is a sample: If $X$ is an
infinite-dimensional reflexive real Banach space and if $T:X\to X^*$ is a
non-monotone, closed, continuous potential operator such that the functional
$x\to \int_0^1T(sx)(x)ds$ is sequentially weakly lower semicontinuous and
$\lim_{\|x\|\to +\infty}(\int_0^1T(sx)(x)ds+\varphi(x))=+\infty$ for all
$\varphi\in X^*$, then the set of all singular points of $T$ is not
$\sigma$-compact.
