Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:

A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{v}(t,x))\dfrac{\partial}{\partial x}\pmb{v}(t,x)+N(x,t)$,

with

$\pmb{v}_i(t_0,x_0)=\pmb{v}_{i0}$, $\forall (t_0,x_0)\in\Gamma_i$, $i=1,...,n$

where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, the $\pmb{v}_{i0}$ are continuous in  initial curves $\Gamma_i$, the initial curves $\Gamma_i$ are non-characteristic and the coefficients of $A$ are constants. In particular, the equations of the system in what I'm really interested have the form:

$0=-\dfrac{\partial b}{\partial x}(t,x)+f_1(x,b(t,x),c(t,x),d(t,x))$,

$0=-\dfrac{\partial c}{\partial x}(t,x)+f_2(x,b(t,x),c(t,x),d(t,x))$

$\dfrac{\partial d}{\partial t}(t,x)=-b(t,x)\dfrac{\partial d}{\partial x}(t,x)+f_3(x,b(t,x),c(t,x),d(t,x))$.

I have only found results for systems in which the matrix $A$ is invertible, but in the case I have, that condition isn't satisfied. So, any suggestion will be really appreciated.