In this paper, the authors consider the Schrödinger-Maxwell type system: $$ \left\{ \alignedat 2 &-{\rm div}(M(x)\nabla u)+a(x)\varphi|u|^{r-2}u=f(x),\quad &&\text{in } \Omega,\\ &-{\rm div}(N(x)\nabla\varphi)=a(x)|u|^r, &&\text{in } \Omega,\\ &u=\varphi =0,&&\text{on } \partial\Omega, \endalignedat \right. \tag1 $$ where $\Omega$ is a bounded, open subset of $\Bbb R^n$, $M$ and $N$ are two uniformly elliptic, bounded matrices, $r>1$, $a(x)\geq 0$ belongs to $L^1(\Omega)$, and $f$ is an $L^1(\Omega)$ function ``controlled'' by $a(x)$ in the sense that there exists $Q>0$ such that $|f(x)|\leq Qa(x)$. \par The authors prove the existence of a solution $(u, \varphi)$ of system $(1)$, in the following cases: \par - if $M=N$ and $M$ is a symmetric matrix. Moreover, in this case, $u, \varphi\in W_0^{2,p}(\Omega)$ and $u\in L^\infty(\Omega)$ (with norm in this space controlled in terms of $Q$), while $\varphi$ is in general not bounded, as a counterexample shows. \par - if $M\neq N$, and in this case, neither $u$ nor $\varphi$ is in $L^\infty(\Omega)$, although $u\in L^{2r+1}(\Omega, a(x))$.