LIE SYMMETRIES AND LINEARIZATION OF THE QRT MAPPING

We study the map u(x)f(2)(u(x+1))]/[f(2)(u(x+1))-u(x)f(3)(u(x+1))], in troduced by Quispel, Roberts and Thompson (QRT). We show, using Lie po int symmetries under what conditions the QRT mapping can be linearised . Requiring that the QRT mapping is invariant under the symmetry vecto r field X(x,u)=alpha(x)partial derivative/partial derivative x+A(x)[BCu+Du(2)]partial derivative/partial derivative u, where B, C and D are constants and alpha(x) is an arbitrary unit periodic function in x, w e derive conditions on the unknown functions f(i) in the QRT mapping. Further for these cases of the QRT mapping we explicitly construct two independent integrals of motion ensuring its integrability. We also d erive its exact solution.