  Under two separate symmetry assumptions, we demonstrate explicitly how the
equations governing a general anti-self-dual conformal structure in four
dimensions can be reduced to the Manakov-Santini system, which determines the
three-dimensional Einstein-Weyl structure on the space of orbits of symmetry.
The two symmetries investigated are a non-null translation and a homothety,
which are previously known to reduce the second heavenly equation to the
Laplace's equation and the hyper-CR system, respectively. Reductions on the
anti-self-dual null-K\"ahler condition are also explored in both cases.
