Extended Hamilton-Jacobi Theory, symmetries and integrability by quadratures

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 $\mathfrak{g}$.