Hi Fabian,
It is not possible to save a search criteria without a value.
I can give you a workaround to achieve the same. The additional search criteria for which you do not want to give any value, you can give the value '* *', i.e. a wildcard, followed by a space, followed by wildcard. This way you will be able to save a search attribute that do not have a concrete value to the saved search. This will act as if you have not given any value for this search field.
Thanks & Best Regards,
Leon