Hamilton-Jacobi Theory: Can we take 2 additive constants?

Question

I am thinking continuously regarding the additive constant in Hamilton-Jacobi theory. But I didn't get a good idea. Why only one additive constant, can we take 2 or 3 additive constants? $$S=S'+\alpha_n+\alpha_{n+1}. $$

Answer 1

A complete solution $$S(q^1,\ldots,q^n,t;\alpha_1,\ldots,\alpha_{n+1})$$ to the Hamilton-Jacobi (HJ) equation should by definition satisfy $$ \det \frac{\partial^2S}{\partial(q,t)\partial\alpha}~\neq~0.\tag{*}$$ If 2 or more $\alpha$ constants enter the solution $S$ in the same manner, e.g. additively, then 2 or more columns in the above matrix would be the same, thereby violating the condition (*).

For more information, see also this & this related Phys.SE posts.