The authors derive the conditions of auto-oscillations of \(\sigma\) (t) in the system \(dx/dt=Ax+b\xi,\) \(\sigma =c^*x\), \(\xi =\phi (\sigma)\), where A, b, c are real constant matrices, \(x\in {\mathbb{R}}^ n\), \(\sigma \in {\mathbb{R}}^{\ell}\), \(\xi \in {\mathbb{R}}^ m\), \(c^*\) is the conjugate matrix to c, \(\phi \in C^ 1\) is a nonlinear vector function. As an example there are given auto-oscillations in biological systems.