Engine stops

The magneto. Most likely the condenser, but could also be the armature windings. If you find either of these issues to be the problem, I strongly reccomed the Brightspark Easycap condensers, as they install on the points plate. This makes for very easy future replacement. I have installed nearly a hunders of them myself & have never had a failure.

In addition, the Brightspark website has an excelent website which covers troubleshooting & repair.

It is also worth noting, that even though I am located in the USA, I use Paul Wolf (from the UK) to wind many of my armatures and not only does he do an awesome job, but he is a club member.

A '59 model would have coil ignition, wouldn't it ?

After 'several hours' the motor itself should have cooled off so ignition components must be suspect.

These intermittent faults are horribly difficult to pin down. I'd be inclined to substitute coil, condensor, spark plug and lead as a matter of course and have a good look at insulation throughout the HT system.

It sounds like it will be either coil or blocked tank vent in cap. Remove coil and get it hot tested on a coil tester,or,put into a warm oven till it feels not too hot to handle. Hook it up to bike and see if it stillstarts.