Can ATS01N2...QN logic inputs LI1, LI2 be switched on using external 24V DC supply?

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Yes, it is possible. Both LI1 and LI2 can be supplied by external 24V DC. Connected following:
0V from external supply to COM terminal on ATS01
24V from external supply via external contacts (push buttons) to LI2 (2-wire control) or to LI1 and LI2 (3-wire control)