Consider a linear ODE system $$\mathbf{y}'=A(x)\mathbf{y}$$ defined on an interval $I$ and $A(x)\mathbf{y}$ satisfying Lipschitz condition for $\mathbf{y}$. As presented in Wiki (the application part), if $A(x)$ is a $2 \times 2$ matrix and we have already known a solution, say $\mathbf{y_0}$, then we can use Liouville formula to determine a fundamental solution matrix.
My question: If $A(x)$ is $3 \times 3$, is there any similar result? Of course two linearly independent solutions $\mathbf{y_1}$ and $\mathbf{y_2}$ can generate a fundamental solution matrix. But can we yield a fundamental solution matrix by knowing only one solution? If not, can we restrict some of the conditions to make it happen? Thanks in advance.