Not quite sure exactly what your question is, but I'll take a stab at it.
There is no such thing as an integrated fork.
The Semi-Integrated, or "Zero Stack" headset standard which has become popular over the past three or four years makes use of the same 1-1/8" fork you'd use with a conventional threadless headset. No difference there. These are typical of the headsets found on Giant, Iron Horse, and a whole host of other brands.
Same goes for truly Integrated headsets, although those are few and far between (Maverick frames come to mind, plus Schwinn's old ICBM headset standard). The fork is the same 1-1/8" fork, but the cupless Integrated isn't exactly a desireable standard to own.
Yeah, I can't tell for sure, but from the eBay photos it looks like a Zero Stack headset. Are you missing parts? Cane Creek and FSA are the two choices for Zero Stack headsets, and both offer very excellent customer-direct support for spare parts and such.
And yeah, any 1-1/8" fork will fit that frame. Just make sure you have all the headset parts.