  In this note we consider the symplectic reduction of a four-dimensional
holomorphic Chern-Simons theory recently introduced in arXiv:1908.02289 for
describing integrable field theories. We work out explicitly the case of the
lambda deformed Principal Chiral Model (PCM) and show that the symplectic
reduction works as a localization mechanism. The reduced Chern-Simons theory
restricts to the set of poles of the twist function underlying the theory,
where the known classical integrability of the lambda deformed PCM can be
reconstructed from the phase space data associated to this set of points in the
spectral space.
