  In this paper we study the one-loop dilatation operator of the full scalar
field sector of Leigh-Strassler deformed N=4 SYM theory. In particular we map
it onto a spin chain and find the parameter values for which the Reshetikhin
integrability criteria are fulfilled. Some years ago Roiban found an integrable
subsector, consisting of two holomorphic scalar fields, corresponding to the
XXZ model. He was pondering about the existence of a subsector which would form
generalisation of that model to an integrable su_q(3) model. Later Berenstein
and Cherkis added one more holomorphic field and showed that the subsector
obtained this way cannot be integrable except for the case when q=e^{i beta},
beta real. In this work we show if we add an anti-holomorphic field to the two
holomorphic ones, we get indeed an integrable su_q(3) subsector. We find it
plausible that a direct generalisation to a su_q(3,2) one-loop sector will
exist, and possibly beyond one-loop.
