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This paper deals with static transfer stability limits (STSL) of network branches and their relation with singularity of the power flow Jacobian. It describes the effect of large transfers on the branch complex flows and shows that when a transfer takes place in the system, at least one branch must reach its STSL before the singularity is encountered. Numerical examples confirm this necessary condition and show that the first STSL appears close to the singularity. Therefore, it is possible to monitor the proximity to the power flow Jacobian singularity by supervising the complex flow in individual transmission lines.