Weighted orbital integrals are distributions on reductive groups over local fields appearing both in the local and global trace formulas. There are associated invariant distributions, which play the same role in the invariant trace formulas. In the case of real groups, the Fourier transforms of these distributions satisfy a system of differential equations.
As a step towards determining those Fourier transforms, we show that this system is holonomic and has a simple singularity at infinity. We deduce that any solution has a series expansion and is a linear combination of certain canonical solutions. For some groups of small rank, we solve the recursion formula for the coefficients explicitly.