A necessary and sufficient condition for nonoscillatory behavior of the solutions of a system of two linear equations of the first order

For the system $y'=b(x)z$, $z'=-a(x)y$, where $a(x),b(x)\in C(x_0,+\infty)$, $b(x)\ge0$ we obtain for $x\ge x_0$ a necessary and sufficient condition for nonoscillatory behavior. From this condition we derive new criteria for the nonoscillatory behavior of the system considered.