Calculation of exotic expansions of solutions to the third Painlevé equation

We consider the exotic asymptotic expansions of solutions to a polynomial ordinary differential equation (ODE). They are such series in integral powers of the independent variable, which coefficients are the Laurent series on imaginary powers of the independent variable. We propose an algorithm for writing ODEs for these coefficients. The first coefficient is a solution of a truncated equation. For some initial equations, it is a Laurent polynomial. Question: will the following coefficients be Laurent polynomials? Here the question is considered for the third Painlevé equation ($P_3$). It appears that for it the second coefficients are Laurent polynomials in all cases, but the third coefficient is a Laurent polynomial under some restriction on parameters.