laplacian

I am working on some boundary integral equation formulation and I am currently stuck with some mathematics. I wish anyone can help me out with this. 

I have an anisotropic (sometimes called generalized) biharmonic differential operator which  takes the form

L = k11 D1^4 +  k12 D1^2 D2^2 + k22 D2^4 

where D1 = d/dx, D2 = d/dy, my problem is two dimensional.

I need to find a fundamental solution (Green's function) for this operator, that is 

L(u) = -delta

where delta is Dirac delta function.