  We consider meromorphic particular solutions of nonlinear ordinary
differential equations and perform a perturbation {\it \`a la} Poincar\'e
making their linearized equation non-Fuchsian at the movable pole and Fuchsian
at infinity. When the nonlinear equation possesses movable logarithms, they are
detected sooner than with the perturbative (Fuchsian) Painlev\'e test.
