  It is shown that any function $G(q_{i}, p_{i}, t)$, defined on the extended
phase space, defines a one-parameter group of canonical transformations which
act on any function $f(q_{i}, t)$, in such a way that if $G$ is a constant of
motion then from a solution of the Hamilton-Jacobi (HJ) equation one obtains a
one-parameter family of solutions of the same HJ equation. It is also shown
that any complete solution of the HJ equation can be obtained in this manner by
means of the transformations generated by $n$ constants of motion in
involution.
