  We study the approaches to two-dimensional integrable field theories via a
six-dimensional(6D) holomorphic Chern-Simons theory defined on twistor space.
Under symmetry reduction, it reduces to a four-dimensional Chern-Simons theory,
while under solving along fibres it leads to four-dimensional(4D) integrable
theory, the anti-self-dual Yang-Mills or its generalizations. From both
four-dimensional theories, various two-dimensional integrable field theories
can be obtained. In this work, we try to investigate several two-dimensional
integrable deformations in this framework. We find that the
$\lambda$-deformation, the rational $\eta$-deformation and the generalized
$\lambda$-deformation can not be realized from 4D integrable model approach,
even though they could be obtained from 4D Chern-Simons theory. The obstacle
stems from the incompatibility between the symmetry reduction and the boundary
conditions. Nevertheless, we show that a coupled theory of
$\lambda$-deformation and the $\eta$-deformation in the trigonometric
description could be obtained from the 6D theory in both ways, by considering
the case that $(3,0)$-form in the 6D theory is allowed to have zeros.
