Polynomial vector fields in ${\mathbb{C}}^{\mathrm{\infty }}$ determining differentiation of hyperelliptic functions of any genus

In this work we give direct proofs of two theorems concerning explicitly defined polynomial vector fields connected to differentiation of hyperelliptic functions of any genus. We prove that the operators determining the fields commute, and we show that each of them annul polynomials defined in terms of generating functions in ${\mathbb{C}}^{\mathrm{\infty }}$.