stay on
(after someone or something) Go to linger on (after someone or something).
stay on
(one's) guard (against someone or something) Go to on (one's) guard (against someone or something).
stay on (some place)
to remain at a place longer than had been planned. I stayed on in Paris for nearly two years. Mary liked it there and decided to stay on.
stay on something
1. to remain on something, such as a horse, road, stool, etc. The first time I rode, I could hardly stay on the horse. It was so slippery that I had trouble staying on the road.
2. to continue to pursue something. I will stay on this problem until it is settled. she stayed on the matter for weeks until it had been dealt with.