  Let us consider, for $n \geq 3$, the Cartan domain
$\mathrm{D}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces
$\mathcal{A}^2_\lambda(\mathrm{D}_n^{\mathrm{IV}})$ we study the problem of the
existence of commutative $C^*$-algebras generated by Toeplitz operators with
special symbols. We focus on the subgroup $\mathrm{SO}(n) \times
\mathrm{SO}(2)$ of biholomorphisms of $\mathrm{D}_n^{\mathrm{IV}}$ that fix the
origin. The $\mathrm{SO}(n) \times \mathrm{SO}(2)$-invariant symbols yield
Toeplitz operators that generate commutative $C^*$-algebras, but commutativity
is lost when we consider symbols invariant under a maximal torus or under
$\mathrm{SO}(2)$. We compute the moment map $\mu^{\mathrm{SO}(2)}$ for the
$\mathrm{SO}(2)$-action on $\mathrm{D}_n^{\mathrm{IV}}$ considered as a
symplectic manifold for the Bergman metric. We prove that the space of symbols
of the form $a = f \circ \mu^{\mathrm{SO}(2)}$, denoted by
$L^\infty(\mathrm{D}_n^{\mathrm{IV}})^{\mu^{\mathrm{SO}(2)}}$, yield Toeplitz
operators that generate commutative $C^*$-algebras. Spectral integral formulas
for these Toeplitz operators are also obtained.
