On a generalisation of the construction of integrable systems determined by symmetric powers of plane algebraic curves

On 17/10/2018, on this seminar V.M.Buchstaber has presented our construction of integrable systems determined by the symmetric power ${\rm Sym }^N (V_g)$ of a plane algebraic curve $V_g\subset \mathbb{C}^{2}$ [bm18]. I have found a simple and useful generalisation of the above construction to the case of $N$–th symmetric power of $V_g\subset\mathbb{C}^{k}$ (for arbitrary $k\in\mathbb{Z}_{>0}$). It also yields new results for symmetric powers ${\rm Sym }^N (V_g)$ of an algebraic curve $V_g\subset \mathbb{C}^{2}$.
[bm18] V. M. Buchstaber and A. V. Mikhailov,  Polynomial integrable Hamiltonian systems on symmetric powers of plane algebraic curves, UMN, December (2018).