Consider $$ u_t + (x-y)u_{x}+yu_{y}-u_{yy}=-xy+x^2+|x|. $$ I figured out that the left hand side is a hypoelliptic operator because the vector fields $A_0:=\partial_{t}+(x-y)\partial_{x}+y\partial_{y}$, $[A_0,A_1]= \partial_{x}-\partial_{y}$ and $A_1:=\partial_{y}$ are linearly independent in $R^{3}$ (triangular with ones on the diagonal). Therefore, solutions would be smooth if the right hand side didn't contain $|x|$. Unfortunately, I don't have a good intuition about the effect of that term.
Could anyone with experience handling such PDEs make an educated guess/prove something about the regularity of solutions?
I am really only interested in the particular term $|x|$, not in $|y|$ or $\|(x,y)\|$.
This is partly a reference request, but please don't link to anything behind a pay-wall, if possible.