I have the 2x2 matrix PDE $$ \partial_t W(x,t) = A(x) W(x,t) + B(x)\partial_x W(x,t) $$ where $W(x,t)=\begin{bmatrix} W_1(x,t), W_2(x,t) \end{bmatrix}^T$ and $A$ and $B$ are 2x2 matrices which depend on $X$. Notably, the matrix $A(x)$ is singular. $B(x)$ is well-behaved and has an inverse.
Were this a simple first order PDE (not a vector PDE), I would proceed via the method of characteristics: $$ \frac{dt}{1} = -\frac{dx}{B(x)} = \frac{dW}{A(x)}. $$ Instead, I am unsure what to do here. How does one find the characteristics of a PDE when the coefficients are matrices?
