You could have a secondary circulating pump that is solar which would just move water through the filter. But you would still need a second pump to run any features on the pool (cleaning system, spa, water features, water circulation, etc.). You would also need the pool equipment very close to the pool, distance & any turns really eat up how effective the pump is at moving water. That added cost of a second pump (along with plumbing), would eat up the electric savings pretty quickly. By the time it paid for itself, it would be about the time to replace it.
A variable speed pool pump uses a lot of electricity at startup to prime and pull the water through the system. Without a battery or large panels, solar will not have enough power for a pool pump at startup or at full speed. At low speed, while circulating the pool, solar could easily do it as a variable speed pump isn't drawing much electricity. Startup, cleaning cycle (running pool vac or infloor), or running a water feature/spa, it will draw a bunch of electricity and is something you do on a regular basis. The pump runs at 3500 rpm at startup and goes down to 2500 rpm when cleaning.
Really, adding solar to the house and using that to power it is the answer. None of the big 3 pool equipment manufacturers (Hayward, Jandy or Pentair) will make such a limited use product like a solar pump. So it would be a DIY setup. Between LED lights and variable speed pumps, pools don't use that much electricity. I haven't measured it, but I have not noticed a significant change in my electric bills since building my pool 2.5 years ago. Everything, except the heater, is ran every day. It cleans overnight (midnight to 6am), the spa spillway runs from 7am until 11am, then the pool circulates from 11am until 2pm. My lights are on from 5pm until 9:30pm and my water features run from 4pm until 8pm. I want to say there is a $30-40 per month difference at most. I spend more on gas heating my spa than I do on electricity.
The biggest savings in electricity that I have noticed is replacing old A/C units. I did that 2.5 years ago and notice a $100 per month (during summer) difference before/after. I realize as the A/C units start to fail, they short cycle (run more often). But it was a dramatic difference.