  We study the effect that the SU(2) symmetry, and the rich Hilbert space
structure that it generates in lattice spin systems, has on the average
entanglement entropy of highly excited eigenstates of local Hamiltonians and of
random pure states. Focusing on the zero total magnetization sector ($J_z=0$)
for different fixed total spin $J$, we argue that the average entanglement
entropy of highly excited eigenstates of quantum-chaotic Hamiltonians and of
random pure states has a leading volume-law term whose coefficient $s_A$
depends on the spin density $j=J/(\mathfrak{j}L)$, with $s_A(j \rightarrow
0)=\ln (2\mathfrak{j}+1)$ and $s_A(j \rightarrow 1)=0$, where $\mathfrak{j}$ is
the microscopic spin. We provide numerical evidence that $s_A$ is smaller in
highly excited eigenstates of integrable interacting Hamiltonians, which lends
support to the expectation that the average eigenstate entanglement entropy can
be used as a diagnostic of quantum chaos and integrability for Hamiltonians
with non-Abelian symmetries. In the context of Hamiltonian eigenstates we
consider spins $\mathfrak{j}=\frac12$ and $1$, while for our calculations based
on random pure states we focus on the spin $\mathfrak{j}=\frac12$ case.
