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This talk will describe my attempts to find an integral for a particular area preserving map. The most important dynamical characteristic of an integrable map is its Fomenko graph and the rotation functions on its edges. I will describe high precision numerics for the computation of the rotation function, with the detection of integrability in mind. For each rational value of the rotation function there exists a periodic orbit. Using discrete reversing symmetries and their fixed sets periodic orbits with large period can be efficiently computed. For the construction of local integrals I will use Birkhoff normal form. The globalisation of such an integral, however, remains difficult.