It should come as no surprise that your eyes may be affected by your cat allergy, too. They're nose-adjacent on your face, after all. These symptoms will often manifest as redness and itchiness. Be sure you don't rub your eyes after coming into contact with a cat and its surroundings, and if your allergic reactions are bad, also wash your hands and lint roll your clothes following any potential petting sessions you may indulge in.
To deal with your symptoms, it can help to keep your distance from the cat in question, perhaps by cordoning it in a particular portion of the house -- or at the very least by keeping it out of key areas like bedrooms and laundry rooms. And remember when your kids promised to take care of the litter box if you agreed to get them a pet? Guess what -- now you have a solid line to rely on when they protest! Same goes for grooming; take it outside, too.