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This is one of those old rumors per say as many people have a different color characteristic and it does not signify a disorder or health condition but is more related to your family history and genes that give us the color of our eyes, hair, and even skn color. If you are not having any health issues then that can be related to your genetic history not a sign of an health condition. When you have regular eye examinations and health examinations this would be something they see and with other symptoms such as headaches, numbness, or tingling it could be a sign of a possible health disorder but even with those symptoms it would be rare that the actual color of your eyes be a health issue. I hope this will help to cause this to not be of worry to you as with all that you state it sounds genetic and not a cause for concern.
You are so very welcome and actually I find rings around the eyes very attractive and it really makes your eyes stand out. I think you and your family are lucky to have this gene trait and I am glad this helps to settle your mind that this is not a cause for worry. Should you ever have any other health concerns please use us again in the future and if you prefer you can request for me as I am on most everyday.