In this talk, we will discuss the critical points and the nodal sets of solutions of Grushin equation. After exactly establishing descriptions about the critical points of the homogeneous Grushin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we obtain that the critical points of solutions of Grushin equation are isolated and each critical point has finite multiplicity. We also show the minimum number of nodal domains of homogeneous Grushin-harmonic polynomials.