  Recently, a variety of deformed $T^{1,1}$ manifolds, with which 2D non-linear
sigma models (NLSMs) are classically integrable, have been presented by
Arutyunov, Bassi and Lacroix (ABL) [arXiv:2010.05573]. We refer to the NLSMs
with the integrable deformed $T^{1,1}$ as the ABL model for brevity. Motivated
by this progress, we consider deriving the ABL model from a 4D Chern-Simons
(CS) theory with a meromorphic one-form with four double poles and six simple
zeros. We specify boundary conditions in the CS theory that give rise to the
ABL model and derive the sigma-model background with target-space metric and
anti-symmetric two-form. Finally, we present two simple examples 1) an
anisotropic $T^{1,1}$ model and 2) a $G/H$ $\lambda$-model. The latter one can
be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev
model.
