Explicitly solvable systems of first-order ordinary differential equations with homogeneous right-hand sides, and their periodic variants

In this paper we identify systems of an arbitrary number $N$ of first-order Ordinary Differential Equations with nonlinear homogeneous right-hand sides of an arbitrary (integer, positive or nonpositive) degree $M$, which feature very simple explicit solutions; as well as variants of these systems—with right-hand sides no more homogeneous—some of which feature periodic solutions. A novelty of these findings is to consider systems characterized by constraints involving their parameters and/or their initial data.