Exotic expansions of solutions to an ordinary differential equation

The exotic expansions in complex powers with constant or polynomials coefficient, for solutions to general ordinary differential equations, are calculated. The complex variable is assumed to range over a universal cover and the real linear function depends on a straight line on the universal cover. Exotic expansions of solutions to ordinary differential equations arise if the characteristic polynomial of the truncated equation corresponding to a vertex has a nonreal root whose real part coincides with the number rk of one of the edges adjacent to his vertex. The expansion also arises if the power solution to the truncated equation has a critical number that is different from the power exponent of this solution but has the same real part at this power exponent.