Families of Invariant Tori in KAM Theory: Interplay of Integer Characteristics

The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts’ attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2.