can this  AXYs be spliced back together and can the controls be fixed?

The pickups look o.k.    I suspect that the electronics may not be Alembic.   It's a little hard to tell because the pics don't give a clear view of the circuits.   I've never seen them use pots with a blue (plastic?) shaft.   Also, it looks like one of you pots has the shaft pulled out. 

Bill, tgo

It seems you have a broken pot, top left.

I think this is a Volume, Pan, Treble and bass set, with -as mentioned above- a broken and replaced pot. Best would be to get two new pots from Alembic I guess.
The bass and treble module connect to each other with the yellow wires, 4 wires straight across.

If you splice the pickup leads back together, you might need to add some shielding over the connection.