Reducibility of 2-Dimensional Ultra-Differentiable Quasi-Periodic Systems with Small Parameters Under Brjuno-Rüssmann Condition

In this article, we delve into whether a two-dimensional ultra-differentiable quasi-periodic system can be reduced. This system is characterized by a coefficient matrix that varies smoothly with a small parameter. We establish, under the adapted Brjuno-Rüssmann non-resonance condition concerning the basic frequencies and without imposing any non-degeneracy assumptions related to the small parameter, that the system can indeed be reduced through an ultra-differentiable quasi-periodic linear transformation for a substantial set of parameters, as perceived in the Lebesgue measure sense.