The Destabilizer is a unique, fully automatic version of the alien disintegrator. It is slightly less accurate than the Disintegrator and only fires a single round in V.A.T.S., despite being automatic. This, coupled with a higher AP cost, makes the Destabilizer a poor weapon for heavy V.A.T.S. usage.
This weapon is found in the weapons lab's shooting range, after beaming up from the space walk and proceeding through the observation deck. It is in the same room as where the Xenotech Expertperk is obtained. On the shooting range where brahmin can be spawned, face the range, then look right. The Destabilizer should be on a low shelf. It is the only item on that shelf. Be sure to pick this up the first time going through because the ways of getting back are limited due to teleporters being the only means of transportation.