I have an operator acting on the polynomial algebra $\mathbb{C}[x,y,z]$ that I would like to find the eigenvalues/eigenvectors of. More specifically, let $P(x_1, \ldots, x_6)$ be a homogeneous polynomial, my operator has the form $P(x,y,z, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$. Are there any general strategies that could help me? For instance, say my operator were: $$z^2y\frac{\partial^3}{\partial x^2 \partial y} + y^3z\frac{\partial^4}{\partial y^2 \partial z^2} + xz^2\frac{\partial^3}{\partial x \partial z^2}.$$ My actual operator is degree 6 and more complicated, but other than that the same type of object. Any thoughts or references on how to attack this type of problem will be very welcome. Thanks a lot!

EDIT: My first example-operator was very poorly chosen, since every monomial would automatically be an eigenvector. I have now altered it a little to avoid this. Keep in mind this was only an example to show what type of object I am considering, and I am looking for general strategies. None the less, thanks for the quick response.

EDIT 2: I had misread the degree of my actual operator - it is 6.