Proof that $M = \left\{(x,y,z): x^2+y^2+3z^3 = xy + 6z^{\frac{1}{3}}, z \neq 0\right\}$ is manifold

Question

Proof that $$M = \left\{(x,y,z): x^2+y^2+3z^3 = xy + 6z^{\frac{1}{3}}, z \neq 0\right\}$$ is differentiable manifold.

The problem is that when $g(x,y,z)=x^2+y^2+3z^3 - xy - 6z^{\frac{1}{3}}$, then $[\partial_x g,\partial_y g,\partial_z g]=[0,0,0]$ in $(0,0,z)$ for some $z$. I do not know how to proceed.