Shots away from the sun are fine, it's only when shooting towards the sun at a slight angle away from it. Is there any way to solve the problem?
Sounds like it's just regular old lens flare. A lens hood is the usual solution, but since the RX100 is a compact camera with no attachment point for a hood, you could improvise some sort of tube, or just hold your hand up near the lens on the sunny side. If your shot actually includes the sun, then there's not a lot you can do to prevent the flare except to reframe the shot or edit the flare out later.
Here's one example of creative use of lens flare on a RX100 from flickr:
Yes, the RX100 suffers from a purple lens flare issue. But then, nearly any camera/lens combination, shooting into a light source, will exhibit flare. You can try shading the lens with your hand, or SteffanB on dpreview recommends opening up the aperture to eliminate the flare: (see this post).
Yeah, I have an RX 100 II, and the purple/blue spot has bit me a couple of times. It's an internal reflection flare that seems to affect the lens in any moderate / strong backlighting situation. It has nothing to do with off-axis light sources, so a lens hood will not help.
Sadly, there really isn't a fix except a different camera. Just be aware that the spot may loom in any backlit scene.