Found out the hard way that I must wear polarized sunglasses above timberline, but did not want to carry two pairs of glasses.
So opted to take only one pair of prescription glasses for trekking, and bought cllip-ons of the type that have hooks that attach to the outer perimeters of the glasses and are spring loaded at the top so the clips hold tightly. Zenni sells them, but only on a couple models now, and you have to buy their glasses also. And Zenni glasses have broken apart on me a couple times. Fine for street wear, but for BP?
I bought better quality polarized clip-ons at an App Mtn Club (AMC) store in NH for under $20, and will now find a frame perhaps on EBay that fits them, and then get lenses to fit the frames from a local optician. Similar clip-ons are also sold at the large truck stops on the cross-country interstates.
If one still needs a case for the one pair when it is not being worn, a semi-hard shell case called the AccuCase is still shown on the web at: https://www.sierratradingpost.com/accucase-wrap-sunglass-case~p~75754/reviews/ The company appears out of biz (they made the cases in the USA), but there are similar ones on EBay, Walmart and Amazon..
Also, the clips on the Zenni clip-ons seem a little dorky, but with the AMC, not so much, as the clips do not project outside the perimeter of the frame.
Of course, one then needs somethings to protect the clip-ons. They are slim and quite sturdy, so a small slip on leather case might do. Depends on your packing style and how the pack is treated.