Air carrier Captain reported encountering mountain wave and severe turbulence during a cimb to cruise altitude resulting in an altitude deviation in an area of known mountain wave activity. The aircraft descended and continued the flight without injuries to the destination.

Narrative

We were flying at Flight level 350 on a ferry flight with 2 person flight crew and no passengers. We were very light so we requested to climb to FL 370 to conserve fuel. ATC said rides were better lower but they had not had anyone try FL 370 so they cleared us to FL 370. After we climbed about a 1000 feet it started to get very turbulent. And then all of a sudden the aircraft hit a lifting action that the autopilot was not able to keep up with even though it captured the altitude at FL 370 it lifted us briefly to about FL 37700. We told ATC that mountain wave and constant moderate turbulence bordering on severe turbulence and we needed to descend back to FL 350. ATC gave us FL 350 back down immediately but it was hard to even descend with the lifting action pushing us back up. Our airspeed was fluctuating a lot and difficult to control. We were near CIM on our route of flight when this happened for location purposes. ATC did ask us about the metric variance. We said very turbulent and mountain wave and not able to control any better. After we descended back to FL 350 constant light to moderate chop for a while thereafter with fluctuations in vertical speed and slight mountain wave. Continued with no further issues