Clean sphere separation with a sphere created from a cube I would like to achieve a clean sphere separation with a sphere created from a cube.

Actually i got some kind of hatched egg effect, but that's not what i need

I was able to achieve that from an UV sphere, but i don't like the pinching effect at the pole :
When textured, i need the sphere to react with lighting effect the same way everywhere. The sphere  will be spinning in a webgl application, and the pinching effect at the pole will be visible.

Is there a way to do it?

I guess i need to add vertices to permit this separation. I tried to play a bit with the knife tool but without success so far.
As Martin Z says, you don't say enough about your problem, also, what problem did you meet with the knife? To cut with the knife, press K for activate, then Z to cut through, then C to cut orthogonally

But if we keep your request, it looks like if you select an edge loop (alt right click), then scale it to 0 on the Z axis with S Z 0, then W > LoopTools > Circle, it works fine. Don't forget to activate the LoopTools addon in the Preferences.

And to separate a mesh in two parts, select the edge loop and press V.