in networkd man page, the search domains are used to handle single-label names:
The domains without the prefix are called "search domains" and are first used as search suffixes for extending single-label host names (host names containing no dots) to become fully qualified domain names (FQDNs). If a single-label host name is resolved on this interface, each of the specified search domains are appended to it in turn, converting it into a fully qualified domain name, until one of them may be successfully resolved.
Both "search" and "routing-only" domains are used for routing of DNS queries: look-ups for host names ending in those domains (hence also single label names, if any "search domains" are listed), are routed to the DNS servers configured for this interface.
I wonder if a single-label name lookup request is handled by LLMNR or by the specified dns severs or both?