Trying to integrate Tutanota's desktop client into the MATE OS turned out to be really quite easy after just a little research.
The process requires three steps;-
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Download AppImageLauncher from https://github.com/TheAssassin/AppImageLauncher/releases/download/v2.2.0/appimagelauncher_2.2.0-travis995.0f91801.bionic_amd64.deb
and open it in Debian Packages Installer (GDebi) where it will be installed in Applications>Accessories.
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Download the Tutanota Appimage from the main page - https://tutanota.com/.
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Open Tutanota's appimage with AppImagelauncher, which will install it in Applications>Internet, and "Robert is your mother's brother".