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In this article we introduce a low order implicit symplectic integratordesigned to follow the Hamiltonian flow as close as possible. This integratoris obtained by the method of Liouvillian forms and does not require particularhypotheses on the Hamiltonian.  The numerical scheme introduced in this paper is a modification of thesymplectic mid-point rule, it is symmetric and it is obtained by an isotopy ofthe deformation of the exact Hamiltonian flow to the straight line passing bytwo consecutive points of the discretized flow. This isotopy generates analternative vector field on the flow lines transversal to the Hamiltonianvector field. We consider only the line arising from the mid-point to constructthe symplectic integrator.

Towards exact symplectic integrators from Liouvillian forms