A note on the existence of vector-functions satisfying differential equations (Q749718)

I. P. Martynoff proved the following result: Theorem \((+)\). If the system \(w'_ k=P_ k(w)\), \(w=(w_ 1,...,w_ n)\), \(1\leq k\leq n\), \('=\frac{d}{dz}\), with homogeneous nonlinear polynomials \(P_ k(w)\), \(k=1,...,n\), has a solution \(w(z)=(w_ 1(z),...,w_ n(z))\) each component of which is an entire transcendental function, then there is a constant vector \(\alpha =(\alpha_ 1,...,\alpha_ n)\), \(\sum^{n}_{j=1}| \alpha_ j| \neq 0\), such that \(P_ k(\alpha)=0\), \(k=1,...,n\). The author proves two theorems, from which Theorem \((+)\) follows as a particular case.