On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations (Q2716886)

The author considers systems of ordinary differential equations with a quadratic homogeneous right-hand side NEWLINE\[NEWLINE\dot x_i=f_i (x_1,x_2, \dots, x_n),\quad f_i= \sum^n_{j,k=1} a_{ijk}x_jx_k. \tag{1}NEWLINE\]NEWLINE To find the values of the parameters \(a_{ijk}\) for which (1) has first integrals, the author gives a new simple proof involving necessary conditions for the existence of polynomial first integrals. It is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials.