  We construct two infinite families of ball quotient compactifications
birational to bielliptic surfaces. For each family, the volume spectrum of the
associated noncompact finite volume ball quotient surfaces is the set of all
positive integral multiples of $\frac{8}{3}\pi^{2}$, i.e., they attain all
possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of
the two families have all $2$-cusps, so that we can saturate the entire volume
spectrum with $2$-cusped manifolds. Finally, we show that the associated neat
lattices have infinite abelianization and finitely generated commutator
subgroup. These appear to be the first known nonuniform lattices in
$\mathrm{PU}(2,1)$, and the first infinite tower, with this property.
