  In this paper, we analyze the 'symmetrized' of the intrinsic Hopf-Lax
semigroup introduced by the author in the context of the intrinsically
Lipschitz sections in the setting of metric spaces. Indeed, in the usual case,
we have that $d(x,y) =d(y,x)$ for any point $x$ and $y$ belong to the metric
space $X$; on the other hand, in our intrinsic context, we have that
$d(f(x),\pi^{-1} (y)) \ne d(f(y),\pi^{-1} (x)),$ for every $x,y \in X$.
Therefore, it is not trivial that we get the same result obtained for the
"classical" intrinsic Hopf-Lax semigroup, i.e., the 'symmetrized' Hopf-Lax
semigroup is a subsolution of Hamilton-Jacobi type equation. Here, an important
observation is that $f$ is just a continuous section of a quotient map $\pi$
and it can not intrinsic Lipschitz.
  However, following Evans, the main result of this note is to show that the
"new" intrinsic Hopf-Lax semigroup satisfies a suitable variational problem
where the functional contained an intrinsic Lagrangian. Hence, we also define
and prove some basic properties of the intrinsic Fenchel-Legendre transform of
this intrinsic Lagrangian that depends on a continuous section of $\pi$.
