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The reason I mention being careful with the lens cleaner idea is that despite the fact that mine worked fine, some have the reputation of being lens destroyers (they use a brush that could potentially scratch the lens). Try and go for a reputable make, not that I know any because I've never really been able to describe the one I got (bought at a local store with few identifying marks on the box).
Good luck!
I had a similar problem with my old PS2, it wouldn't read my GT4 disc properly and it actually scratched and ruined two copies of it.
Answer? Buy a slimline PS2.
The disk is probably damaging the PS2. The reading lens may have to be working harder to read the disk.
It doesn't sound as if the PS2 is damaging the disk, because other disks work fine.
Answer? Get a new GTA:SA disk...