Answer: Wish I could provide a list of dog friendly ground cover plants, but there are very few. Ivy is the toughest but it can be terribly invasive so may not be a good choice in the long run. The damage done by dog urine on lawns and other ornamental plants is caused by excess nitrogen in one spot, and that "burns" plants.
If you could spray water consistently every time wherever there is dog urine on the grass, the problem would go away, and the diluted nitrogen would actually green up your lawn.
Best wishes with your lawn!
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