gramatická analýza: ... bručel dr. Mejzlík, chytající se stébla
Present active participle from chytat se (3pl. chytají se plus -cí)
The main verb (the conjugated verb) is bručel; the subject of this sentence is therefore 3sg masc., dr. Mejzlík (...mumbled dr. Mejzlíik who was trying to catch straw, mumbled dr. Mejzlík, trying to catch (any possible) straw) This is the only noun that the PrAP can modify. The PrAP is thus in the nominative case, masc. sg.
Stébla is in the genitive since chytat se requires the geniitve
as its object.
Chytat se stébla is a fixed expression reporting that the person was in desparate need of something and would welcom any kind of help, the paraphrase may also be slightly different from a simple který construction, involving nejvyšší nouze (greatest, highest need), vítat pomoc (welcome help)