in the white paper you have 2 sides : attacker and honest node What does it mean that there is either the attacker or honest node. 
This means that within 10 minutes either attacker or honest node mines a block but what about neither OR another side ? How could this be interpreted ?

Does it make sense or what is interpretation of neither attacker nor honest node succeed in finding a block ? What would that mean for the consensus rule that there is a third side ?
This is open to interpretation, but this is how I see it.

Honest Nodes

Accept the blockchain history and do not try to retroactively modify it.

Attacker Nodes

Nodes that try to change data in past blocks and recalculate the proof of work. Note:


  To modify a past block, an attacker would have to redo the
  proof-of-work of the block and all blocks after it and then catch up
  with and surpass the work of the honest nodes. See Bitcoin Whitepaper pg. 3


51% Attack

The reason this is important is because, theoretically if more than half of the nodes on the network are trying to attack the blockchain, they might succeed and effectively gain control because collectively, they can compute the proof of work faster.


  Does it make sense or what is interpretation of neither attacker nor
  honest node succeed in finding a block ? What would that mean for the
  consensus rule that there is a third side ?


I'm not sure what exactly you would define as a "third side", but if, for example, there was a third group that was not "honest" it would be considered another attacker. Now, if the third group is not attacking the blockchain in the same way as the other attacker group, that could split the chain into 3 different forks: 1 "honest" chain, and 2 different "compromised" chains.