Let $K$ be a field of characteristic $0$, let $A = K[[t_1, \ldots, t_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes_A \Omega^1_{A/K} = \bigoplus_{k=1}^n V \ dt_k$ be an integrable connection, so $\nabla^2 = 0$. I want to show that there always exist solutions $s \in V$ to the equation $\nabla(s) = 0$ subject to any initial condition $s(0) = \alpha$ for $\alpha \in V(0) = V \otimes_A A/\mathfrak{m}_A$.

More concretely, letting $e_1, \ldots, e_r$ be a basis for $V$, I have a system of $n \times r$ linear first-order homogeneous partial differential equations $$ \frac{\partial}{\partial t_k} s_i = \sum_j M^j_{ik} s_j $$ For some $M_{ik}^j \in A$. Or, writing $M_k$ for the matrix $(M_{i,k}^j)_{i,j}$, $$ \frac{\partial}{\partial t_k} s = M_k s $$

How can I show that integrability implies there are always solutions? Concretely, integrability is exactly the statement that for all $i,j,k, \ell$, we have $$ \frac{\partial}{\partial t_\ell} M_{k} - \frac{\partial}{\partial t_k} M_{\ell} = M_k M_\ell - M_\ell M_k $$