  We give an explicit formula for the motivic integrals related to the Milnor
number over spaces of parametrised arcs on the plane with fixed tangency orders
with the axis. These integrals are rational functions of the parameters and the
class of the affine line. Using a set of natural recurrence relations between
them, we prove an unexpected invariance property with respect to the
simultaneous inversion of the parameters and the class of the affine line. We
also discuss a generalization of this system of recurrence relations whose
solutions are also symmetric and satisfy additional differential equations.
