  The purpose of this paper is to study the existence of weak solutions for
some classes of one-parameter subelliptic gradient-type systems involving a
Sobolev-Hardy potential defined on an unbounded domain $\Omega_\psi$ of the
Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ ($n\geq 1$) whose
geometrical profile is determined by two real positive functions $\psi_1$ and
$\psi_2$ that are bounded on bounded sets. The treated problems have a
variational structure and thanks to this, we are able to prove the existence of
an open interval $\Lambda\subset (0,\infty)$ such that, for every parameter
$\lambda\in \Lambda$, the system has at least two nontrivial symmetric weak
solutions that are uniformly bounded with respect to the Sobolev
$HW^{1,2}_0$-norm. Moreover, the existence is stable under certain small
subcritical perturbations of the nonlinear term. The main proof, crucially
based on the Palais principle of symmetric criticality, is obtained by
developing a group-theoretical procedure on the unitary group
$\mathbb{U}(n)=U(n)\times\{1\}$ and by exploiting some compactness embedding
results into Lebesgue spaces, recently proved for suitable
$\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space
$HW^{1,2}_0(\Omega_\psi)$. A key ingredient for our variational approach is a
very general min-max argument valid for sufficiently smooth functionals defined
on reflexive Banach spaces.
