UNDER CONSTRUCTION

The engine is the really "big ticket" item when it comes to building a gyro and it is natural to want to save some money in this area. Unfortunately, there are not a lot of viable options beyond the basic 447 or 503 Rotax. In order to safely fly the aircraft, you will need the following with respect to engine/performance parameters:
Rotax
The suitability of Rotax 447 and 503 engines is not in question. If a new engine costs too much, consider a used engine with the investment in a complete overhaul. The result can be a like-new engine at a considerable savings. The advantages of using a Rotax engine are not trivial:
Hirth
There are several Hirth engines in the 40-50 hp range that could do as well or better than the Rotax engines. I consider many aspects of the engineering a fabrication of Hirth engines to be superior to Rotax, which may help to explain their somewhat higher cost. That said, there has been a significant reliability issue when using Hirth engines on gyros. The company had a tendency to push the use of free-air models, which often resulted in overheating and siezed pistons. Use of fan-cooled models may change this picture to a great degree. Service and parts availability will be restricted, compared to Rotax and you would need to check with Hirth owners to see if that is an issue.
Kawasaki
Kawasaki 440 engines (nominally rated at 40 hp) are available from some sources at VERY attarctive prices. The implication is that you can have a working engine for $500-600. Hardly! Most folks will spend $1500 to $1800 to get a working engine with all the "extras" needed to fly. The extras include a belt-drive reduction system for the prop. Unfortunately, we had the opportunity to try a 440 on the prototype Honeybee and could not get more that 130-160 pounds of static thrust out of the system. That engine never flew the aircraft out of ground effect. The same aircraft flew very well on a new Rotax 447. The engine prove reliable enough and it was easy to start and ran well. It just doesn't seem to run well-enough to fly a gyro (it would be great for a single-seat trike). It is possible you could work up an effective engine, but it won't be plug-and-paly and it will cost a lot more than you anticipate. There is no way the parts and service options can equal those of Rotax.
Half-VW
Some half-VW variants are rated around 40 hp and look attractive, especially since they are four-strokes. Unfortunately, they are direct-drive and cannot generate the needed torque. Any use of these engines would require a PSRU, significantly increasing weight and requring proper matching of the prop. Not likely!
Motorcycle Engines
Bike engines are typically light, four-stroke, and have significant power output. This has caused lots of folks to look at them as a four-stroke alternative to engines like the Rotax. You can typically forget most Japanese engines since the engine case and transmission are one unit. That said, I know of one guy who used a Yamaha engine on a fixed wing, using the transmission as the PSRU (he flew in second gear, as I remember). You can remove the transmission from BMW engines, so what about one of these as an aero-engine? Well, apart from the huge problems I will get to in the next section, there is the heat issue. A gyro engine has to operate continuously at 75% (cruise) or greater power output for most of its lifetime. This duty cycle is far higher than a typical motorcycle engine can handle! As result, the engine will overheat. Bike engines cannot dump the heat load incurred in flight service. If it was easy to use bike engines on aircraft, it would already have been done.
Parting Thoughts
Adapting other engines for flight seems easy - put together
or modify a PSRU, fiddle with the prop and gear/belt ratios and the job
is done. Nothing could be further from the truth:
The single biggest factor that will strike the balance
between worry and comfort in flying a gyro is the reliability of the engine/drive
system. If, some evening, you are facing a nasty engine-out landing, you
may wonder about the money you saved by using an alternate engine!