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 in the link you can find the system: System of pde

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.

*Edit, sorry but for some reason the editor was not allowing me to write the matrices that I need in a good way, so I put a link to the image of the system that I create with latex.