@culzean - it sounds logical, but if it is a cable, then why the car starts with pushing. After that if I stop it, it starts normally. I also tried just to push the car without trying to start it. It does not start with the key again, situation does not change.
@ColinS -
1. it never happened while driving forwards, only backwards, but still it could be a coincidence
2. When I rotate the key it shuts down everything, sometimes you can't even hear a click. When I get it back power is restored.
3. It seem like accidental fuel in the pistons, but as far as I read, it is very hard to achieve that with injection engine systems
4. A friend of mine had similar problem, a small part of his starter motor was missing brushes. The car works, but when you stop it, if the motor stops in the position where are the missing brushes, no current, no rotation , no start. But in my case it never happened on cold start. Only when parking.