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When in check and only move exists can be automatically made?

Sorry, conditional moves are only possible in Correspondence chess. However, since your opponent has a tendency to stop moving in lost positions, maybe lichess.org/report is worth a try.
that happened to me...one move to mate and the guy stops moving. i reported it. no idea if that does anything. people stopping moving when they are losing, or sure to lose, or about to get checkmated, is pretty common. i just surf the web until the clock runs out.
I am not talking about conditional but un-conditional move, forced.
That's totally not the point, dear @Toadofsky.
It is perfectly possible that Cris7 missed many good moves. But in this situation it is clear what he demands.

I usually say: well, you know the risk when starting a game. When the opponent stops moving in a lost position, you never can lose more time than you initially agreed to spend on this game.

But with only one legal move ...

Still, it would be bewildering, @Cris7. Imagine the opponent would also have only one move ... so the engine has to make these moves without time for either player?

And for playing chess: The system is probably not programmed to "know" if there are several legal moves. It suppresses illegal moves (like ignoring check), and it recognizes checkmate and stalemate. But I think calculating if there are at least 2 legal moves might require ressources that are better spent on other things.
@Sybotes : Well, the system detects mate in less than 1 second. The calculation about only 1 legal move should be made in 1 minute at most, because as I mentioned, my opponent did not move for the rest 8 minutes at last move. Then a message could pop-up: "You have only one legal move: h4xg5" and if no response after 1 minute, that move be made.
Even offline you have the choice between moving, resigning, moving plus draw-offer.

No, it is consistent. Otherwise forced moves would be executed after 1 min „mid-game“.
I like the idea of adding an "announce mate" feature where if you announce the incorrect number of moves, you lose. But I don't think developers would accept code contributions for that.
"Otherwise forced moves would be executed after 1 min „mid-game“. Better so than wait many minutes an opponent who does not move anymore.

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