  We introduce a method based on auxiliary master equation for solving the
problem of an impurity with local electron-electron and electron-phonon
interaction embedded between two conduction leads with a finite bias voltage.
The Anderson-Holstein Hamiltonian is transformed to a corresponding Lindblad
equation with a reduced set of sites providing an optimal approximation of the
hybridization function. The problem is solved in the superfermion
representation using configuration interaction for fermions and bosons. The
phonon basis is shifted and rotated with the intention of permitting a low
phonon basis cutoff even in the strong-coupling regime. We benchmark this
approach with the numerical renormalization group in equilibrium, finding
excellent agreement. We observe, however, that the rotation brings no advantage
beyond the bare shift. This is even more apparent out of equilibrium, where
issues in convergence with respect to the size of the phononic Hilbert space
occur only in the rotated basis. As an application of the method, we explore
the evolution of the phononic peak in the differential conductance spectra with
changing phonon frequency.
