Nevertheless I heard (I should investigate this rumour) that being lazy enabled an easier integration of the IO system so there seems to be a connection between pure (i.e. side-effect free) and laziness.

Perhaps you're thinking of the observation that laziness essentially forced Haskell to be pure (otherwise the order of side effects is very difficult to control) and thus led to the invention of monadic IO.

If you're thinking of something else please do report back if you investigate as I would be interested to know.

They chose to make Haskell a lazy language and that forced them to make it pure. You cannot have laziness without purity.

You can, but you'll really hate it :)