The BME singles--the 44 & 50--are Echo chainsaws. The 44cc engine is from a CS-44, so I would guess the 50 is from the CS-50. They sell the 44 short blocks separate at lawn & power shops, but they are nearly $300. By the time you add the carb, ignition and prop hub, you are close to the cost of a BME, but without the warranty.
If you want to convert a cheap engine, go with the Poulan 46. It's very close to the same weight and power, but can often be found on Ebay for under $150. I've got a whole page on Engine Conversions....
I've weighed my BME 44 and Brison 2.4 side by side, but don't remember the exact numbers. I do, however, remember that the BME was 5 ounces lighter. The Brison 3.2 is 1/2-3/4 pound heavier still. Kieth says the 50 is only 3oz heavier than the 44.
Cliff
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