Electrical – Light bulb burning problem on a two lights luminaire

NO PUTTY!

You have a loose electrical connection and a loose mechanical connection.

Mechanical

First you need to fix the mechanical attachment mechanism. It may be a loose nut on a threaded pipe holding the individual fixture to the base. It could be some other locking mechanism, such as bolts or screws. In some fixtures, the sockets are held in by pressure fits, which are harder to repair.

In either case, you probably need to take at least the lighting unit off of the fan, and maybe the whole fan down, depending on how it is put together. Before you do anything, turn off the circuit to the fan, not just the wall switch or the remote control switch. Once the wiring is exposed, double check that all wires are not hot by using a non-contact tester.

You need to restore the original attachment method, not just a glue or putty to wedge it in place. Fan vibration will almost certainly cause a makeshift repair to break again.

Electrical

It sounds as if you have a loose wire. You need to make sure that all wires, hot neutral, and ground are all connected (in the US, these are black or red, white, and green or bare). The ground may be established by physical contacts among the metal parts, and restoring the mechanical connection may fix an intermittent ground. Sometimes a wire may be under a wire nut, but not well connected. If there is no obvious disconnected wire, remove and check wire connections under wire nuts.

Replacement

If you cannot make the physical socket stable as originally designed, you need to replace the fixture (or at least that part of it). Underwriters approval (a safety standard) is dependent on the structure as designed. If you cannot restore the mechanical structure, you probably need to replace the entire fan.

From your description it sounds like you have the intelligence to understand this stuff, but are simply lacking certain nuggets of information. You'll want to read up a lot on "switch loops" and particularly "3-way switches". It will all make sense pretty quickly, then.

Oh, and one more thing that's a bit harder to uncover: In America, wire colors do not have firm meanings. Green always means ground. White or gray always mean neutral, unless marked or in switch loops and messengers - the goal being to build the necessary circuits with readily available multi-wire cable even though it's the wrong color. "Hot" lines are everything else - including marked whites and (implied) switch-loop whites, and there is no official standard of how to designate certain types of hot. (however in commercial facilities, consistency is required.)