Suppose in a private network setup , one node runs a standard geth client, and then we add another node to the network using a modified geth, where only the difficulty has been made static. Can this node be added successfully as a peer to the existing node in the network ? If yes , how will the consensus algorithm work if a differential difficulty exists in the network (I don't think its possible , but have doubts) . If no , what are the exact reasons for such a set up not working ?
It is not going to work. The difficulty is a part of the
Hash of the block, and if your node has modified the difficulty, the hash will be different and its block won't be accepted by the other nodes in the network.
you can check the difficulty function in
consensus/ethash/consensus.go file in go-ethereum source code