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This thread is resolved. Here is a description of the problem and solution.
Problem: How to set language switcher in homepage
Solution: Generate menus at Appearance -> Menus and translate them. Enable the menu language switcher which can be activated at WPML -> Languages -> Menu Language Switcher. The language switcher option will only be available, if there is currently a translation for the content available.
Relevant Documentation: More information about this topic you can find here: https://wpml.org/documentation/getting-started-guide/language-setup/language-switcher-options/