\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?