  In this paper, we study the extended Hamilton-Jacobi Theory in the context of
dynamical systems with symmetries. Given an action of a Lie group $G$ on a
manifold $M$ and a $G$-invariant vector field $X$ on $M$, we construct complete
solutions of the Hamilton-Jacobi equation (HJE) related to $X$ (and a given
fibration on $M$). We do that along each open subset $U\subseteq M$ such that
$\pi\left(U\right)$ has a manifold structure and
$\pi_{\left|U\right.}:U\rightarrow\pi\left(U\right)$, the restriction to $U$ of
the canonical projection $\pi:M\rightarrow M/G$, is a surjective submersion. If
$X_{\left|U\right.}$ is not vertical with respect to $\pi_{\left|U\right.}$, we
show that such complete solutions solve the "reconstruction equations" related
to $X_{\left|U\right.}$ and $G$, i.e., the equations that enable us to write
the integral curves of $X_{\left|U\right.}$ in terms of those of its projection
on $\pi\left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical,
we show that such complete solutions can be used to construct (around some
points of $U$) the integral curves of $X_{\left|U\right.}$ up to quadratures.
To do that we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$
of $G$, an explicit expression up to quadratures of the exponential curve
$\exp\left(\xi\,t\right)$, different to that appearing in the literature for
matrix Lie groups. In the case of compact and of semisimple Lie groups, we show
that such expression of $\exp\left(\xi\,t\right)$ is valid for all $\xi$ inside
an open dense subset of $\mathfrak{g}$.
