There is no way to sync a catalog between desktop and laptop. You would have to keep the catalog on an external drive and connect it to each machine.
I would recommend using a separate travel catlog on the laptop and then import as a catlog to combine with your main catalog.
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There is no Sync between two different desktop system. The sync you may be referring to is between one install of LR desktop and one catalog file on that one computer and the LR Mobile app for iOS and Android. Even then it just syncs collections between the 2 system and not the complete catalog. That is unless you put all your images into one collection.
LR is basically a Single computer program. Although Adobe allows you to install on 2 different computers so you can have it on a desktop and a notebook you use for travel and when away from your primary system those 2 installs can not be run concurrently on both system.
The Catalog file must be stored on a hard drive directly connected to the system LR is running on. It can't be stored on a networked drive.
Thank you!!! unfortunately you confirmed my expectations.
Thank you for your help