  We prove that to every connected Abelian subgroup $H$ of the biholomorphisms
of the unit ball $\mathbb{B}^n$ we can associate a set of bounded symbols whose
corresponding Toeplitz operators generate a commutative $C^*$-algebra on every
weighted Bergman space. These symbols are of the form $a(z) = f(\mu^H(z))$,
where $\mu^H$ is the moment map for the action of $H$ on $\mathbb{B}^n$. We
show that, for this construction, if $H$ is a maximal Abelian subgroup, then
the symbols introduced are precisely the $H$-invariant symbols. We provide the
explicit computation of moment maps to obtain special sets of symbols described
in terms of coordinates. In particular, it is proved that our symbol sets have
as particular cases all symbol sets from the current literature that yield
Toeplitz operators generating commutative $C^*$-algebras on all weighted
Bergman spaces on the unit ball $\mathbb{B}^n$. Furthermore, we exhibit
examples that show that some of the symbol sets introduced in this work have
not been considered before. Finally, several explicit formulas for the
corresponding spectra of the Toeplitz operators are presented. These include
spectral integral expressions that simplify the known formulas for maximal
Abelian subgroups for the unit ball.
