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