Given a $1$ form $\theta$ on $\mathbb{R}^3$ find a 2-dimensional submanifold $M$ such that $\theta|_{M}=0$

Question

I am asked to find a $2-$dimensional submaifold of $\mathbb{R}^3$ such that the $1$ form $$\theta=(z^2+2xy)\cdot dx+(x^2+2yz)\cdot dy+(y^2+2zx)\cdot dx$$ restriced to the manifold $M$ equals zero. My idea is to find $M$ trough its tangent space. Since we know that for $p\in M$ and arbitraty $V_p=a(\frac{\partial}{\partial x})+b(\frac{\partial}{\partial y})+a(\frac{\partial}{\partial z})\in T_p M$ that $$\theta V_p=a(z^2+2xy)+b(x^2+2yz)+c(y^2+2zx)=0.$$ But I feel like I don't know enough to construct $M$ from this equation.

Answer 1

Hint: I agree with @studiosus that probably your formula is wrong and you wanted to have $(y^2+2xz)dz$ as the last summand. If this is the case, try writing $\theta$ as $df$ for a smooth function $f$ and then the level sets of $f$ will do the job.