Hello and thanks for trusting me to help you and your Tuly today.
From what you describe of her symptoms and her history, I am most suspicious of a low grade chronic herpes virus related eye infection. It is likely that she was exposed to the virus at some point in her life. Most cats are and so this is not abnormal. Most cats, once exposed tend to carry the virus in a latent form in the nasal tissue and it can flare up intermittently on and off throughout their life. When it does flare up, it tends to be in a mild form and may only affect the eyes. Since the eyes are connected to the nose through the tear ducts, when her eyes are producing more discharge, it is possible that it will show up as the dark line you are describing.
Remember that this is usually viral in nature and that is why the antibiotic probably had little to no effect. Antibiotic eye drops are sometimes used if a secondary infection develops (noted by a yellow or green eye discharge), but in general, her immune system is the only thing that will help the signs to resolve. As she develops an immune response, she can get the virus back in check (not get rid of it, but back to not showing symptoms). If she is not vaccinated for herpes, I would recommend having that done.
The vaccine is not designed to prevent the virus, but to lessen the symptoms of it does become symptomatic again. The best thing you can do to keep everyone feeling well is to keep them stress free and otherwise as healthy as they can be. Once they become stressed, due to environment or other health issues, this is when the latent virus is likely to flare up and lead to symptoms.
There are products to strengthen their immune system, like interferon and lysine (an amino acid) but these are usually used when a pet is having symptoms that they can't seem to shake.
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