Do you have access to a gym? When I was out with a shoulder injury I did a lot of stationary biking and leg machines (like thigh presses and calf raises etc). You can also do ab stuff like dish (http://www.crossfitignitesydney.com.au/fitness-and-training-tips/a-short-analysis-of-the-dish-hold) with your injured arm resting on your stomach/chest, depending where the burn is.

Well, if you can’t really move, standing and sitting around sounds like a perfect idea!
This guy teaches some of the basic stances of Taekwondo, https://www.youtube.com/watch?v=uZYhVAmZ7N0 (couldn’t find a pretty lass on such short notice) and while he shows really great Love to the basic stances and walking, I’d recommend the one he describes starting at minute 12. The horse seat stance is something that needs to be practiced ad absurdum and is a firm staple of several martial arts. It does wonders for your posture, back and works out your legs. If you’re feeling too comfortable while holding it, because you have Chuck Norris’ quads, try lowering yourself deeper into the stance, or raise one of your legs or both up on your tippy toes.See how comfortable your arms are at your sides, if they’re not very comfortable because of the burns, stretch them out straight ahead of you or to the sides, or if you have a sling, mirror your wrapped arm’s position with the healthy one .Just be symmetrical.
Pay attention to detail and balance and most of all posture!
Always try to do stances barefoot!
And breathe deeply.
Maybe even ask some of the trainers at your gym if they have martial arts experiece and can help you with that.
I know stances aren’t the coolest of all martial arts aspects, but they build core strength beyond the muscular. I can’t really explain their uh, magical properties, but they’re like workout and mediation in one.
If I haven’t managed to rope you into some basic Taekwondo, just try sitting against a wall,entire back, flat aganst the wall, mind you, with your upper and lower legs at a ninety degree angle and hate me after ten seconds.
Cross one leg over the other, raise your feet onto your toes, for some variation, whatever hurts best:-) Find the pain. Repeat.
I mentioned the barefoot thing, because now is the time to show some love to the muscles in the soles of your feet. They are often in rather bad shape due to underuse, but they really make for good and flexible stances and posture and those really help your back out a ton!
Stand on one leg while brushing your teeth. Shift your weight onto the balls of your feet whenever standing around somewhere and unlock your knees, practice dynamic standing.
Do heel raises, whenever possible, switch, and raise your toes and walk on your heels a bit! (careful with the sides of your feet! Don’t do that for now) Also if you’re taking Ciprofloxacin, or have been taking Ciprofloxacin for a while, do not do the heel walking! Long term use of Cipro can lead to spontaneous Achilles tendon rupture, and nobody needs that)
Whenever the obvious opts out for a while, it’s an opportunity to find the neglected and smaller things, like you know, your feet.
I hope you find something that works for you!Take Care!