Consider the following degenerate PDE: $$\label{eq:eq1}\tag{*} u_{xx}+2u_{xy}+u_{yy}-u=0\quad \text{ in }\mathbb{R}^2 $$
The PDE is not uniformly elliptic. Let us however define $v$ as the solution to $$v_{zz}-v=0\quad \text{ in }\mathbb{R}$$ which is a uniformly elliptic equation. Furthermore, defining $$u(x,y):=\frac{1}{4}v(x+y)$$ we find a solution to \eqref{eq:eq1}.
I am thus wondering about the hopefully precisely enough posed question: If the PDE is degenerate over the whole domain, can I embed it into a lower dimensional space where it is in this case uniformly elliptic to exploit regularity results.
If that is the case, what are the references to understand this, also for more complicated PDEs?
Yes, you are correct. But keep in mind that the obtained regularity is with respect to the variables describing your lower-dimensional manifold. Your solutions may still be very irregular in the "orthogonal" direction.