The HJ equation is a non-linear first-order PDE for $S$ in $(n+1)$ variables $(q^1, \ldots, q^n, t)$, but the PDE does not depend on $S$ directly, only its derivatives. Therefore one additive integration constant $S\to S+\alpha_{n+1}$ is trivial.

The HJ equation is a non-linear first-order PDE for $S$ in $(n+1)$ variables $(q^1, \ldots, q^n, t)$, but the PDE does not depend on $S$ directly, only its derivatives. Therefore one additive integration constant $S\to S+\alpha_{n+1}$ is trivial.

For more information, see also this related Phys.SE post.