Yeah I would agree with Rabelais that it could be the result of contact irritation. Something you may put directly onto your lips, like your chapstick. Since you mentioned it flares up, have you ever noticed anything in common when the flares up happen? Maybe something you ate, something in your daily skin care routine? If you have been trying a lot of products on your lips, that can also be overwhelming your skin. It really just looks like really dry lips to me that are peeling, kind of like eczema without the red rash.
Have you tried washing with sea salt? Especially dead sea salt, is very mineral rich so it’s actually very moisturizing, can help exfoliate the dead skin off and it’s a natural antibacterial and contains just basically sea salt minerals and water, so the ingredient list is simple. I would probably dissolve about 1-2 tbsp in a 1/2 cup of warm water. You can add more or less if you want, but keep in mind that not only the more salt you use the more it will sting, but some people actually are sensitive to the high magnesium content in Dead Sea Salt and it itches/burns. But the latter is not a concern with regular sea salt. You can either dip cotton balls or a clean cloth like a J-cloth and hold it to your lips. Keep it soaked for a few minutes or you can even try up to 30mins-1hr. I use this for my eczema, and then I just wash it off and don’t apply anything else on my face for the day.