Are there any general methods to solve linear coupled multivariate partial differential equations of the first order?

For an example:

Are these coupled system of differential equations of two variables solvable?

\begin{align} a \psi(x,y)+ (b x-g) \partial_{x}\phi(x,y)+(cy-g) \partial_{y}\phi(x,y)-\left(m+gx\right)\phi(x,y)+\left(m+gy\right)\phi(x,y)&=0\\ a \phi(x,y)+(bx+g) \partial_{x}\psi(x,y)+(by+g) \partial_{y}\psi(x,y)-\left(m-gx\right)\psi(x,y)-\left(m-gy\right)\psi(x,y)&=0 \end{align}

Here $a,b,c,g$ and $m$ are constants.

Any hint toward a solution and also any references for these kinds of equations would be really helpful.