In this paper, starting from the combination of two in volutive systems, we consider separately some systems of polynomial functions:{Im(1)=Bm Rm}m=0∞, {Im(2)=Bm Sm}m=0∞, {Im(3)=Bm Tm}m=0∞ and so on; and analyse carefully the sufft cient and necessary conditions of the involution of three systems of functions {Im(i)}m=0∞(1≤i≤3) with general coefficients.Furthermore,we present concrete forms of the involutive systems hidden in {Im(i)}m=0∞(1≤i≤3);and thus obtain six kinds of nontrivial involutive systems of functions, which include a few involutive systems discussed inthe literature. Based upon these involutive systems, we can generate a lot of new finite-dimensional Hamiltonian systems which are completely integrable in the sense of Liouville. |