  Context. Slowly pulsating B (SPB) stars display multi-periodic variability in
the gravito-inertial mode regime with indications of non-linear resonances
between modes. Several have undergone asteroseismic modeling in the past few
years to infer their internal properties in a linear setting. Rotation is
typically included in the modeling by means of the traditional approximation of
rotation (TAR).
  Aims. We aim to extend the set of tools available for asteroseismology, by
describing time-independent (stationary) resonant non-linear coupling among
three gravito-inertial modes within the TAR. Such coupling offers the
opportunity to use mode amplitude ratios in the asteroseismic modeling process,
instead of only relying on frequencies of linear eigenmodes.
  Methods. Following observational detections, we derive expressions for the
resonant stationary non-linear coupling between three gravito-inertial modes in
rotating stars. We assess selection rules and stability domains for stationary
solutions and predict non-linear frequencies and amplitude ratio observables
that can be compared with their observed counterparts.
  Results. The non-linear frequency shifts of stationary couplings are
negligible compared to typical frequency errors derived from observations. The
theoretically predicted amplitude ratios of combination frequencies match with
some of their observational counterparts in the SPB targets. Other observed
ratios could be linked to other saturation mechanisms, to interactions between
different modes, or to different opacity gradients in the driving zone.
  Conclusions. Our non-linear mode coupling formalism can explain some of the
stationary amplitude ratios of observed resonant mode couplings in single SPB
stars monitored during 4 years by Kepler.
