Three of my watch-fixer friends were Accutron specialists from the 1960s up until they started disappearing from wrists. All three used the same method to clean dirty index wheels on old 'Rattytrons' without taking them apart (me too):
Run a small drop of hairspring cleaner (One Dip etc) down a needle or small oiler onto the index wheel a time or two to rinse the dirt away (do not touch the index wheel teeth if at all possible). If it hums and just will not run, this may get it going. If it does not hum, check the batt (making sure it is 'upside down'), contacts, coils, and circuit.
If the watch runs up until the date is changing and stalls out but the second hand still moves (218), it is usually the canon pinion...turn it over. How? remove the hands and dial, then remove the hour wheel and look at it...easy to see how to do it. The 218 quick sets the date, many do not know it and this is why so many canon pinions are loose.
Do Not turn the sweep second hand backwards with a brush, tool etc when working around the watch with the movement out of the case or the crystal removed, it might/will bend the index and pawl jewel springs (the springs were supposedly made from hairspring material). Do not leave a running 218 'hacked' for long.
If you do much work on them, you will probably need a microscope...and lay off coffee and energy drinks.
A regular 214 can be turned into a pretty good Spaceview using Clark's crystal and hands. Sometimes you will need a 'skeletonized' minute chapter ring though.
I learned enough about A-trons to get by but will not fool with one now unless I have to.
Trivia:
Remember what Crocodile Dundee said: "That 218 is not an Accutron. This 214 is an Accutron!"
Don Johnson wore a steel 214 Spaceview on 'Nash Bridges'.
http://members.iinet.net.au/~fotoplot/acc.htm