  Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version
of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" -
\Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" -
\Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases}
$$ describing the equilibria of an elastically-coupled extensible double-beam
system subject to evenly compressive axial loads. Necessary and sufficient
conditions in order to have nontrivial solutions are established, and their
explicit closed-form expressions are found. In particular, the solutions are
shown to exhibit at most three nonvanishing Fourier modes. In spite of the
symmetry of the system, nonsymmetric solutions appear, as well as solutions for
which the elastic energy fails to be evenly distributed. Such a feature turns
out to be of some relevance in the analysis of the longterm dynamics, for it
may lead up to nonsymmetric energy exchanges between the two beams, mimicking
the transition from vertical to torsional oscillations.
