  We study the $(n-1)$-dimensional volume of central hyperplane sections of the
$n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an
alternative, simpler argument for proving that the volume of the section
perpendicular to the main diagonal of the cube is strictly locally maximal for
every $n \geq 4$, which was shown before by L. Pournin. Then, we prove that
non-diagonal critical central sections of $Q_n$ exist in all dimensions at
least $4$. The crux of both proofs is an estimate on the rate of decay of the
Laplace-P\'olya integral $J_n(r) = \int_{-\infty}^\infty \mathrm{sinc}^n t
\cdot \cos (rt) \mathrm{d} t$ that is achieved by combinatorial means. This
also yields improved bounds for Eulerian numbers of the first kind.
