Always have ruts and holes develop in lawn due to ongoing activities and life in general.
You and I may differ in what constitutes 'lawn'; should call ours a yard probably--more like improved burmuda pasture than golf course.
Grass cutting is by spouse on high-speed ZTR that doesn't like (spouse or machine) ruts holes mounds.
Assume you are in custody of loader or rear scoop / drag.
Order out a 10-wheeler of topsoil and put it somplace out of the way on the place. These can be the first ruts you fill. Avoid underground lines if soft or shallow and remember septic and laterals.
Wait till hard dry then pick up a load with spouse on machine and you on shovel and fill ruts holes. Unless you're talking 'holes' in which case you need to order more material.
Backdrag with loader till smooth. Otherwise and in any case fine tune with heavy hand-rake. We use 'asphalt' rake not standard lightweight 'bow-back' yard rake. Fertilize, seed (sprig, sod), water.
We pick up 6-ft landscape rake and go over the place once a year or after wind- or icestorms. Keeps everything mostly level and picks up an amazing amount of detritus and debris.
Regarding landscape rake: have a 3-point conversion attachment that fits quick-attach loader. Allows use of 3-point implements on front. Operator can drive backward and see precisely up-front where tines are and apply up-down pressure and angle as needed. Handy tool. Also useful with gin pole, disk, 6-way blade, anything unpowered. Careful with rake as can bend tines if not cautious. Handiest cleaning ditches with blade or rake.
We're in long hard drought. If similar and have pond and it's dry go get a bucket from it to use for rut fill. A little silt and clay won't hurt and the price is right while cleaning pond.
Ordered load of 'form sand' [used forming up concrete work; locally mined not river] to mix with pond clay for better quality yard fill material and discovered cost and effort spent mixing wasn't worth additional improvement over pond clay alone.
Sidelight: owned dump truck for a while. Discovered total expense compared to payback wasn't worthwhile. Fuel, insurance, maintenance, time spent in line at quarry, round-tripping all equals no benefit approaching only break-even not counting time. Much better to pay trucker couple times a year.
Post back and let us know how you proceed and results. Before-after photos good too.