I know that $x^3-xz^2-y^2z=0$ is a nonsingular curve in $\mathbb{P}^2$(char $k\neq 2)$. By definition, each local ring is a regular local ring. Consider the ring of degree zero elements in a homogeneous localized ring $(k[x,y,z]/(x^3-xz^2-y^2z))_{(x,z)}$.
Therefore, this ring should be a regular local ring. But I don't know how to find the generator of the principal maximal ideal. I just know it can be $(x/y, z/y)$, and we have the relation $$z/y=(x/y)^3-(x/y)(z/y)^2.$$
One can write $z(xz+y^2)=x^3$ in the quotient, so $(z/y)(xz/y^2+1)=(x/y)^3$ in the homogeneous localization, so that $(z/y)$ is already in the ideal generated by $(x/y)$ since $(xz/y^2+1)$ is a unit.