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We prove that to every connected Abelian subgroup $H$ of the biholomorphismsof the unit ball $\mathbb{B}^n$ we can associate a set of bounded symbols whosecorresponding Toeplitz operators generate a commutative $C^*$-algebra on everyweighted 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 $\mathbb{B}^n$. Weshow that, for this construction, if $H$ is a maximal Abelian subgroup, thenthe symbols introduced are precisely the $H$-invariant symbols. We provide theexplicit computation of moment maps to obtain special sets of symbols describedin terms of coordinates. In particular, it is proved that our symbol sets haveas particular cases all symbol sets from the current literature that yieldToeplitz operators generating commutative $C^*$-algebras on all weightedBergman spaces on the unit ball $\mathbb{B}^n$. Furthermore, we exhibitexamples that show that some of the symbol sets introduced in this work havenot been considered before. Finally, several explicit formulas for thecorresponding spectra of the Toeplitz operators are presented. These includespectral integral expressions that simplify the known formulas for maximalAbelian subgroups for the unit ball.

Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball