My amenorrhea lasted three months plus the time to get a doc appointment and then for the Rx to take effect--so round up to 4 months. I don't recall any side effects other than what I'd normally experience with PMS, but I'm sure it's different for everyone. You could check with a pharmacist about possible side effects.
Honestly my cycle still isn't as regular as it was before my amenorrhea last year, but I think that's an issue with me since it's been a year now since I took the drug. I doubt it's still lingering in my system.
Regarding whether low body fat percentage can lead to amenorrhea or not, I have no personal experience, since I was never at a BF low enough for anyone to consider that as a possibility. I do, however, think dietary fat is an issue. At this time last year I was undereating, then went on a 1200-cal "diet" that did have me eating more cals than I had previously--but it cut down dietary fat. So the decrease in dietary fat (I stupidly cut out coconut and more importantly cod liver oil) could have been a factor in the amenorrhea as well.
And for whether it's necessary for you to get your period back, well, if it were me and knowing I'm trying to get back to a state of health and that a female body "wants" to be in a state where it could have a baby and all that, I guess my take would be that getting a period would be a sign that you're back where you ought to be. That's me, though. But hopefully Leigh will see and chime in, as well. I think she has more knowledge about this than I do.
