I think your eyes get burning may because they don't have 100% UV protection or they are not dark enough.You can try polarized sunglasses which can reduce glare so that your eyes can be protected from harmful lights. There is no neccessary for you to get name brand sunglasses. Go to choose polarized sunglasses that have 100% UV protection, then your eyes won't get burning.
You should add 100% anti-UV radiation coatings or choose the polarized lenses which are prevent much glares. It is the lenses on the sunglasses effect much instead of the frames. Ran Ban would be your best choice, for it's comfort, price and large selection of fashion frames.