An oil stream falling vertically into a stationary bath of the same liquid, if its flow speed is above a critical threshold, because of its viscous property, will induce a transition where a “cusp” on the free surface is “cracked” open by the jet. This industrially important phenomenon has been well studied. However, results from the present study indicate that there is still some puzzling issue of this phenomenon, concerning the stability of the “cusp” before the transition. And this issue becomes more complicated when a constant bath motion is introduced. New “dynamical states” that are not possible when the bath is stationary can arise when energy is continuously supplied into the system by the externally driven constant flow of the bath liquid. Multiple “dynamical states” can exist for the same experimental conditions. The system can exhibit slight hysteretic behavior.