Depends what the lamp and lampholder are made off, to decide if needs earth.
Is the wire hanging from the lampholder.
If the holder is metal there may be an earth terminal inside the holders base, if not you may have to get an english holder with an earth terminal fitted.
Some lamps only need 2 core but they are always double /insulated and have no metal parts to become hazardous.
ONLY IN THIS CASE can the earth be left off.
Needs checking as the earth May Be Required,as said fit a 3 amp fuse also.
If you mean the wire is by the LAMP holder , and that is made of plastic .
You only will need to earth inside one of the holders and the earth may then pick up through out all the metal.
If the earth wire is elsewhere or near the threaded bar you can get an earth tag ,like a washer but with a terminal on it, to fit on the bar and nut up.
Get from a lighting shop and connect the earth on to that.
After check with multimeter continuity from plug earth pin to the metal parts.
Depending how the shade fits, if it has one.
And the way it fits on the metal bar, usually 10mm or 1/2 inch threads are the norm.
you could change the lamp holder's for brass one's with an earth terminal inside it , when connected the earth will then earth the holder , metal bar and the fitting if all are screwed together,but cannot say 100% without seeing it as the existing design may not require earthing.
You can also get plastic " continental lampholders " these have metal thread inside with an earth tag the earth will then earth the metal of the lamp only.
The lampshade fitter is bigger on these though.
May be safer to take to a lighting shop and get done if your in doubt.
I have read and re-read your post. And I can't quite understand where the earth wire is. Some part of the metalwork should be earthed. I would expect if any part is earthed then it would link to all other parts. It is one of those things that we could see at a glance but takes pages to write about. We do have special tools to test but I would think if it looks right it most likely is right.
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