We investigate the numerical implementation of a symplectic integrator combined with a rotation (as in the case of an elongated rotating primary). We show that a straightforward implementation of the rotation as a matrix multiplication destroys the conservative property of the global integrator, due to roundoff errors. According to Blank et al. (1997), there exists a KAM-like theorem for twist maps, where the angle of rotation is a function of the radius. This theorem proves the existence of invariant tori which confine the orbit and prevent shifts in radius. We replace the rotation by a twist map or a combination of shears that display the same kind of behaviour and show that we are able not only to recover the conservative properties of the rotation, but also make it more efficient in term of computing time. Next we test the shear combination together with symplectic integrator of order 2, 4, and 6 on a Keplerian orbit. The resulting integrator is conservative down to the roundoff errors. No linear drift of the energy remains, only a divergence as the square root of the number of iterations is to be seen, as in a random walk. We finally test the three symplectic integrators on a real case problem of the orbit of a satellite around an elongated irregular fast rotating primary. We compare these integrators to the well-known general purpose, self-adaptative Bulirsch--Stoer integrator. The sixth order symplectic integrator is more accurate and faster than the Bulirsch--Stoer integrator. The second- and fourth- order integrators are faster, but of interest only when extreme speed is mandatory.