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GCU Grey Area uncovered the following information for us.
The rare Mk3 Particle Repeater is Mk3 is offered in exchange for a "Spacefly Egg Collection", which is crafted from 5 "Spacefly Eggs (authentic)" items. In order to find the latter, think about Spacefly egg lore 
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Note that other items with Spacefly in the name exist so take care to be looking for "Spacefly Eggs (authentic)".
Mission details
After returning to the NPC offering the mission with a crafted collection, you can either exchange it for the opportunity to purchase a Particle Repeater Mk3 or sell it for profitsss.