68 a-body left indicator pulses

strange,,all the lights-directionals-stop lights,park lights work great but the left directional indicator pulses when the engine is running,and it doesn't increase or decrease with engine speed,grounded cluster,dask,engine to firewall,neg termanal to core support,its got me wondering about the bulkhead through the firewall??????????????

Depending on year...If a turn light fails...your indicator light stays on!

My cluster indicator stayed on when my left turn light "ground wire" fell off the light housing itself. Perhaps your wires are worn or have a loose connection. I suggest you recheck both the front and rear grounds closely and by hand. A engines vibration and loose connection is enought to make your indicator pulse and screw with your head!
PS Also check the bulb sockets to ensure there's no bulb play.

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