Some classes of analytic potentials with second-order poles and corresponding asymptotic solutions of the Schrodinger equation are studied. The main goal of the analysis is to answer the question as to whether a Stokes phenomenon and therefore a reflection takes place for those equations. It is shown that one can notice the Stokes phenomenon not only in the leading term of an asymptotic expansion but also in higher-order ones. Therefore for only a restricted set of potentials proposed, the reflection coefficient is exactly equal to zero although it looks like that for other potentials in the first approximation. This letter gives rise to a further extension of the original approach of Berry and Howls (1990).