  Proton emission from deformed nuclei is described within the non-adiabatic
weak coupling model which takes into account the coupling to
$\gamma$-vibrations around the axially-symmetric shape. The coupled equations
are derived within the Gamow state formalism. A new method, based on the
combination of the R-matrix theory and the oscillator expansion technique, is
introduced that allows for a substantial increase of the number of coupled
channels. As an example, we study the deformed proton emitter $^{141}$Ho.
