In which cases the lift distribution is not elliptical? (i.e. e≠1)
When is the lift distribution not elliptical?


The distribution is non elliptical basically always, you have to struggle quite a lot to achieve an elliptical distribution.

As far as I know only the Spitfire managed to decently approximate the elliptical distribution, and even then, the unavoidable presence of the fuselage had its consequences.


  What are the physical or ideal causes?


Ideal? I think you are confusing what is ideal here. The ideal case is the elliptical distribution, since it gives you the minimal induced drag for equivalent lift.

As basic theory goes, you can have an elliptical distribution with an elliptical wing (is a consequence of the so-called lifting-line theory).