I think the 'eyes' were Royal Enfields answer to the night-time parking requirements that England had (has?) following WW II.
A lot of Brit bikes used a small bulb inside the headlight and some of the 7" headlights available for the RE still have them. That's what the small gray wire inside the Casquette is for.
Digging on the web I found these British laws concering parking at night. As far as I know, they still apply (although many don't follow them).
Parking at night
248
You MUST NOT park on a road at night facing against the direction of the traffic flow unless in a recognised parking space.
[Laws CUR reg 101 & RVLR reg 24]
249
All vehicles MUST display parking lights when parked on a road or a lay-by on a road with a speed limit greater than 30 mph (48 km/h).
[Law RVLR reg 24]
250
Cars, goods vehicles not exceeding 1525 kg unladen weight, invalid carriages, motorcycles and pedal cycles may be parked without lights on a road (or lay-by) with a speed limit of 30 mph (48 km/h) or less if they are
at least 10 metres (32 feet) away from any junction, close to the kerb and facing in the direction of the traffic flow
in a recognised parking place or lay-by
Other vehicles and trailers, and all vehicles with projecting loads, MUST NOT be left on a road at night without lights.