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We obtain through a Matrix Product Ansatz (MPA) the exact solution of themost general $N$-state spin chain with $U(1)^N$ symmetry and nearest neighbourinteraction. In the case N=6 this model contain as a special case theintegrable SO(6) spin chain related to the one loop mixing matrix for anomalousdimensions in ${\cal N} = 4$ SYM, dual to type $IIB$ string theory in thegeneralised Lunin-Maldacena backgrounds. This MPA is construct by a map betweenscalar fields and abstract operators that satisfy an appropriate associativealgebra. We analyses the Yang-Baxter equation in the N=3 sector and theconsistence of the algebraic relations among the matrices defining the MPA andfind a new class of exactly integrable model unknown up to now.

The Matrix Product Ansatz for integrable U(1)^N models in Lunin-Maldacena backgrounds