Lie symmetries and linearisation of the QRT mapping

We study the map u(x + 2) = [ƒ1(u(x + 1)) − u(x)ƒ2(u(x + 1))]/[ƒ2(u(x + 1)) − u(x) ƒ3(u(x) + 1))], introduced by Quispel, Roberts and Thompson (QRT). We show, using Lie point symmetries under what conditions the QRT mapping can be linearised. Requiring that the QRT mapping is invariant under the symmetry vector field X(x, u) = α(x)∂/∂x+A(x)[B+Cu+Du2]∂/∂u, where B, C and D are constants and α(x) is an arbitrary unit periodic function in x, we derive conditions on the unknown functions ƒi 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 derive its exact solution.