Is it possible to run two nodes with the same public key? Is it possible to run two nodes with the same public key on the Lightning network? The reason is reliability -- two different servers run different instances of node software with the same setup (incl keys) -- if one fails -- the other one continue serving. Possible?

thank you.
With reliability, I assume you mean protecting channel balances against server failure (Not being able to close the channel correctly following an error).

You can reproduce the wallet in both nodes, but that doesn't contribute to securing a payment channel's balance in the case of a failure. It simply just means both node instances have identical wallet keys.

In other words, the two node instances may be signing with the same wallet keys, but they cannot be operating as a single peer for a given channel. 

Since it is the individual payment channel balance (unbroadcasted, signed commitment transactions) which is at risk, it would be necessary to secure the commitment transaction data of a single node instance to protect your balance in case of a server failure.