CE-560 Captain reported a runway excursion onto the adjacent grass after landing at night at a non-towered airport. The Captain was able to maneuver back onto the runway with rudder control and taxied off the runway without damage.

Narrative

This was the fourth flight leg of the day; none of which were over one hour. Both crewmembers were well rested; had eaten supper before this flight and had not had flight duties the day before the occurrence. Total flight time for the day was 3.3 hours.We were on a night approach to ZZZ at the end of a flight leg of about 42 minutes; from ZZZ1. The approach had been briefed before the start of the flight and reviewed during the short cruise portion of the flight.The airport was visible from at greater than 5 miles out. The surrounding area is very dark. We were flying a visual approach while using the RNAV (GPS) RWY XX approach for back-up to the visual. Wind was missing from the airport weather; but the winds at nearby stations was light and variable. Both pilots confirmed we were on course; on the proper descent path; needles centered; and on approach and ref speeds. Runway lights were illuminated using 5 radio clicks; while some distance from the airport and again on final approach.Both pilots believed that we touched down on the center-line of the runway; just prior to the end of the 1000 ft; fixed distance markers. The landing was smooth; but shortly after the nose-wheel touched down the aircraft veered left and partially onto the grass to the left of the runway. I was able to get the aircraft back on the runway with rudder pedal control and completed the roll-out and taxi to parking without further issues. In debriefing the event with the other pilot; we were both quite sure the aircraft had touched down on the centerline of the runway. On short final; we had verbally confirmed that we were on the glidepath and on the extended centerline of the runway. Despite that; the review of the tire marks and tracks in the grass the next day (I was not a part of that review) apparently showed the aircraft touched down on the left edge striping of the runway and probably veered left when the nosewheel touched down on the edge of the pavement of the runway. The aircraft landing lights were on; the runway lights were on medium; which is the setting I generally prefer at night; and the cockpit and cabin lights were set at an appropriate level for night flight.To date; neither pilot can understand how we were not on the runway centerline for touchdown; assuming the review of the tire-tracks is correct.I have a recurrent training coming up; scheduled well before this event occurred; and plan to request that we recreate this approach in the simulator.