Bulbs too high wattage?
The 3AJ originally came with 35W/35W bulbs in each of its headlights. If they are changed for the more common 60W/55W bulb the alternator can't cope and the battery will run down. It would be worth checking if a previous owner has made this mistake, If so, the simple solution is to just unplug one of the bulbs.
Hope it is this simple for you. If not, you need to get hold of a multimeter and check the charging circuit. When one of the three windings went open circuit on my 3AJ it ran fine except if I put my (single 60/55W bulb) headlight on, when the batery went flat over a few hours of riding.
let me know if you need advice on checking for this once you have discounted the bulbs.