I have a Mexican thornless lime and have this same issue. Mine is caused by aphids which are mined by the Argentine ants. The aphids secretion causes the sooty mould.
As it is only about five feet tall, I spray with Safer insectical soap on a regular basis. I do not have the same problem with my lemon or satsuma. I have often wondered why the lime is more susceptible to aphid infestation.
What systemic would you use?
