  A search for Beyond the Standard Model double-$\beta$ decay modes of
$^{76}$Ge has been performed with data collected during the Phase II of the
GERmanium Detector Array (GERDA) experiment, located at Laboratori Nazionali
del Gran Sasso of INFN (Italy). Improved limits on the decays involving
Majorons have been obtained, compared to previous experiments with $^{76}$Ge,
with half-life values on the order of 10$^{23}$ yr. For the first time with
$^{76}$Ge, limits on Lorentz invariance violation effects in double-$\beta$
decay have been obtained. The isotropic coefficient
$\mathring{a}_\text{of}^{(3)}$, which embeds Lorentz violation in
double-$\beta$ decay, has been constrained at the order of $10^{-6}$ GeV. We
also set the first experimental limits on the search for light exotic fermions
in double-$\beta$ decay, including sterile neutrinos.
