  In this paper we aim at characterizing the gauge balls in the Heisenberg
group $\mathbb{H}^n$ as the only domains where suitable overdetermined problems
of Serrin type can be solved. We discuss a one parameter family of
overdetermined problems where both the source functions and the Neumann-like
data are non-constant and they are related to the geometry of the underlying
setting. The uniqueness results are established in the class of domains in
$\mathbb{H}^n$ having partial symmetries of cylindrical type for any $n\geq 1$,
and they are sharper in the lowest dimensional cases of $\mathbb{H}^1$ and
$\mathbb{H}^2$ where we can respectively treat domains with $S^1$ and
$S^1\times S^1$ invariances.
