  Besides its various applications in string and D-brane physics, the
$\theta$-deformation of space (-time) coordinates (naively called the
noncommutativity of coordinates), based on the $\star$-product, behaves as a
more general framework providing more mathematical and physical informations
about the associated system. Similarly to the Gelfand-Dickey framework of
pseudo differential operators, the Moyal $\theta$-deformation applied to
physical problems makes the study more systematic. Using these facts as well as
the backgrounds of Moyal momentum algebra introduced in previous works [21, 25,
26], we look for the important task of studying integrability in the
$\theta$-deformation framework. The main focus is on the $\theta$-deformation
version of the Lax representation of two principal examples: the $sl_2$
KdV$_{\theta}$ equation and the Moyal $\theta$-version of the Burgers systems.
Important properties are presented.
