  A search for neutrinoless $\beta\beta$ decay processes accompanied with
Majoron emission has been performed using data collected during Phase I of the
GERmanium Detector Array (GERDA) experiment at the Laboratori Nazionali del
Gran Sasso of INFN (Italy). Processes with spectral indices n = 1, 2, 3, 7 were
searched for. No signals were found and lower limits of the order of 10$^{23}$
yr on their half-lives were derived, yielding substantially improved results
compared to previous experiments with $^{76}$Ge. A new result for the half-life
of the neutrino-accompanied $\beta\beta$ decay of $^{76}$Ge with significantly
reduced uncertainties is also given, resulting in $T^{2\nu}_{1/2} = (1.926 \pm
0.095)\cdot10^{21}$ yr.
