  Several integration schemes exits to solve the equations of motion of the
$N$-body problem. The Lie-integration method is based on the idea to solve
ordinary differential equations with Lie-series. In the 1980s this method was
applied for the $N$-body problem by giving the recurrence formula for the
calculation of the Lie-terms. The aim of this works is to present the
recurrence formulae for the linearized equations of motion of $N$-body systems.
We prove a lemma which greatly simplifies the derivation of the recurrence
formulae for the linearized equations if the recurrence formulae for the
equations of motions are known. The Lie-integrator is compared with other
well-known methods. The optimal step size and order of the Lie-integrator are
calculated. It is shown that a fine-tuned Lie-integrator can be 30%-40% faster
than other integration methods.
