Your insurance will pay for that surgery, but only under certain circumstances, for example if it's an emergency.
under certain circumstances
In certain situations. I let my kids sleep with me in my bedroom under certain circumstance, like if they've had a nightmare.
under certain circumstances
and under certain conditions
Fig. depending on or influenced by something; because of something. Under certain conditions, you can see across the lake to the other side. Under certain circumstances, what you propose to do is all right.