Nonuniqueness of conjugate flows

The problem of weakly stratified flows conjugate to a uniform flow with a prescribed density distribution over its depth is considered. The sufficient condition of existence and uniqueness of the conjugate flow is obtained for a smooth generic background density profile. If the condition obtained is violated, it is shown that the number of branches of conjugate flows and their asymptotic behavior near the bifurcation point are determined by the fine structure of stratification.