car shuts off when lights turn on
 

Hello. It is my first time here so please excuse me if i will not do something right.
Background: 1997 Honda Accord EX 2.2. VTEC Station Wagon 220K

Problem: the car starts and drives perfectly fine, however when I turn my headlights on the engine usually starts to fluctuate or even shuts off completely. So I need to switch on the lights before I start the car, after what the car drives just fine. recently the problem started to occur during the time when i blink with my high beams.

The car was converted aftermarket xenon lights(stock optic) 6000K.(better visibility- reason for installation) Tried to replace the xenon back with stock bulbs the problems still exist.
New Bosch battery
Alternator shows 14V constantly -/+ 0.5V

Question: What it could be? I suspect that it is ignition switch( electrical part of it). It was recalled some time ago so I will try to go to the dealer tomorrow and find out whether it was replaced by the warranty or not. But still I have doubts. Relay? Fuse box? wiring? Alternator?
Please share your experience.
Thanks.

Are you measuring this voltage at the battery or the "charging post" of the alt. Also when you are testing at the post, do this while someone turns on the lights and then blinkers-heck turn on the ac and set the blower to high. Does it still hold?

And should have asked this first, and stored codes in the ECU?

Sorry for late reply.
just a quick update on my problem.
today installed new alternator and checked battery in Advance Auto. The battery is like new condition according to them(it is less than year old and i never had a problem with starting a car).
voltage still the same 14V +/- 0.5V
Still problem have a problem. the problem is intermittent and usually occurs only at the first time i switch the headlights. sometimes however what is interesting the car comes to near a complete shut off and the check engine light appears on the dash. I restart the car-it is gone. No codes in ECU.
So what i can try next? I plan to go to the junk and pick up headlight relay and dimmer relay and try them. But at this point i have doubts that it will help. another question could it be that Integrated Control Unit???
please respond with ideas.