Does this P.D.E have analytic solutions?

[Kplex]

I would like to know whether this P.D.E is solvable and if yes, for which values of a.

2 (y Cos[a] + x Sin[a]) - 4 (y Sin[a] - x Cos[a]) f[x, y]
- 2 (y Cos[a] + x Sin[a]) (f[x, y])^2
+ ((x^2) Cos[a] - 2 x y Sin[a] - (y^2) Cos[a]) D[f[x, y], x]
- ((x^2) Sin[a] + 2 x y Cos[a] - (y^2) Sin[a]) D[f[x, y], y] == 0

I have tried to solve it for the case where a=0
2 y + 4 x f[x,y] - 2 y f[x,y]^2 - 2 x y (f^(0,1))[x,y] + (x^2-y^2) (f^(1,0))[x,y]==0,
on MATHEMATICA but after 3 hours it gave me no result.

Then I utilised the method of characteristics and expressed the a=0 case as a system of three ODES
dx/dt = (x^2 - y^2)
dy/dt = (-2xy)
dw/dt = 2yf^2 - 2y - 4xf
but MATHEMATICA still gave me no results. So I suspect that this PDE might not have an analytic solution.
Is this PDE solvable, and if not could you please indicate a reference which indicates which cases of PDES are non solvable? I am only interested in Analytic solutions.

Any kind of help will be most appreciated.
Thank you in advance.

 

[gtoguy97]

Unfortunately, this PDE does not have an analytic solution in general. It is possible to solve it for certain specific values of a, such as a = 0, but it cannot be solved for arbitrary values of a. There are no references that indicate which cases of PDEs are nonsolvable, since each PDE needs to be solved separately depending on its specific form and parameters.