The paper considers a system of three differential equations of the following form: \begin{align*} \dot x_1&=a_{11}x_1+\varphi_2(x_2)+\varphi_3(x_3), \\ \dot x_2&=a_{21}x_1+a_{22}x_2+a_{23}x_3, \\ \dot x_3&=a_{31}x_1+a_{32}x_2+a_{33}x_3. \end{align*} It is established that the given system can have periodic solutions if the generalized Routh–Hurwitz conditions hold.