First integrals and exact solutions of a system of ordinary differential equations with power nonlinearity (Q2405328)

Consider systems of ordinary differential equations of the type \[ {dx_k\over dt}= S_k x^{\lambda_k}_k+ x^{\mu_k}_k\,\sum_{j\neq k} a_{kj} x^{\nu_j}_j,\quad 1\leq k\leq n,\leqno{(*)} \] where \(S_k\) and \(a_{kj}\) are real numbers, \(\lambda_k\), \(\mu_k\), \(\nu_k\) are rational numbers. The authors derive conditions on these numbers such that an explicit solution of \((*)\) can be given.