I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even problems to find some references. I would be really great to have some advice about how to face the problem. What I would like to do is of course find a solution of this system but even proving the existence of the solution can be a first great step. So thank you in advance for any possible help and some good reference. Here is the system:

$\begin{pmatrix} 0 & f_1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & f_1 \end{pmatrix}\partial_x \begin{pmatrix} N_1 \\ N_2 \\ N_3 \end{pmatrix} + \begin{pmatrix} f_2 & 0 & 0 \\ 0 & 0 & f_2 \\ 0 & 0 & 0 \end{pmatrix}\partial_y\begin{\pmatrix} N_1 \\ N_2 \\ N_3 \end{pmatrix} + begin{pmatrix} 0 & 0 & 0 \\ 0 & f_3 & 0 \\ f_3 & 0 & 0 \end{pmatrix}\partial_z \begin{pmatrix} N_1 \\ N_2 \\ N_3 \end{pmatrix}= \begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3\end{pmatrix}$

Where we have three indipendent variables $(x,y,z)$ and three dependent variables $(N_1,N_2,N_3)$. The functions $\omega_i:R^3\rightarrow R$ $f_i: R^3\rightarrow R $, $N_i:R^3\rightarrow R $ are all smooth functions. Given that the matrix of coefficients are all singular I don't know how to proceed even about the classification of this system. Like I said I would be interested in a local solution or at least the proof of the existence of the solution. Any reference or help about this system will be really appreciated.