  In this letter, we derive the Korteweg-de Vries (KdV) equation corresponding
to the surface dynamics of a shallow depth ($h$) two-dimensional fluid with odd
viscosity ($\nu_o$) subject to gravity ($g$) in the long wavelength weakly
nonlinear limit. In the long wavelength limit, the odd viscosity term plays the
role of surface tension albeit with opposite signs for the right and left
movers. We show that there exists two regimes with a sharp transition point
within the applicability of the KdV dynamics, which we refer to as weak
$(|\nu_o|< \sqrt{gh^3}/6)$ and strong $(|\nu_o|> \sqrt{gh^3}/6)$
parity-breaking regimes. While the `weak' parity breaking regime results in
minor qualitative differences in the soliton amplitude and velocity between the
right and left movers, the `strong' parity breaking regime on the contrary
results in solitons of depression (negative amplitude) in one of the chiral
sectors.
