This thread is resolved. Here is a description of the problem and solution.
Problem: The main menu on my website is not fully synchronized with the menu of the secondary language. As a result, when I try to change the language the menu does not change and remains in its original language (not expected), the language of the page changes (expected), and the flag disappears (not expected). Secondary Menu was created manually instead of translating it.
Solution: The pages were set not to be translatable in WPML → Settings → Post Type Translation and this was making the language switcher point to the same language page.
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