Picture from the motherboard around the battery.Agree with nglevin on the Astec's having issues.
Inspect UK2 and UL2 for any corroded/damaged traces (right next to the battery). The shutdown circuit for the IIcx sits right next to a bunch of caps so it's likely that there's a damaged trace if the PSU turns out to be good.
The PSUs are exchangeable.

I'm sorry - I find it hard to believe if it was done by professional, but this re-cap job is rather messy - if it was a professional would've cleaned the board 100% and straighten the caps out.The board recapped by a friend who is professional.As for power switch I tried everything.Still nothing...