On the Allen-Cahn equation in the Grushin plane: a monotone entire   solution that is not one-dimensional
  We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows. 
Isabeau Birindelli, Enrico Valdinoci