  In this paper we prove existence of least energy nodal solutions for the
Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u =
|x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in }
\Omega, \qquad u=v=0 { on } \partial \Omega, \] where $\Omega$ is a bounded
smooth domain in $\mathbb{R}^N$, $N\geq 1$, $\alpha, \beta \geq 0$ and the
nonlinearities are superlinear and subcritical, namely \[ 1>
\frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}. \] When $\Omega$ is either a ball
or an annulus centred at the origin and $N \geq 2$, we show that these
solutions display the so-called foliated Schwarz symmetry. It is natural to
conjecture that these solutions are not radially symmetric. We provide such a
symmetry breaking in a range of parameters where the solutions of the system
behave like the solutions of a single equation. Our results on the above system
are new even in the case of the Lane-Emden system (i.e. without weights). As
far as we know, this is the first paper that contains results about least
energy nodal solutions for strongly coupled elliptic systems and their symmetry
properties.
