Compound predicate is a great structure of two verbs which in combination convey more complicated meaning that if they would be separately. Moreover, if we deal with a compound predicate, we can't divide it into parts because its meaning will be changed. Every verb fulfills its own function and complies with certain rules in the frame of a compound predicate. The lexical meaning is expressed with a main verb (indefinite form - infinitive), the grammatical one - with auxiliary one (we can decline this verb). Here is the example of declension of a compound predicate.
Он начал (что делать?) танцевать - He started (doing what?) dancing. Он мечтал (что делать?) танцевать - He wanted to (do what?) dance. Он боялся (что делать?) танцевать - He was afraid of (doing what?) dancing.
The point of a compound predicate is that you cannot use the half of it not losing the meaning. Besides, the action of a verb in infinitive form is always directed to a subject - this is one of conditions of forming a compound predicate. If infinitive is referred to another part of speech, you cannot say that infinitive is a part of a compound predicate.
The main meanings of auxiliary part of a compound predicate are the phase (beginning, continuation, ending) and modality (necessity, ability, evaluation).
Phase Он бросил курить - He quitted smoking. Она начала готовиться к экзамену - She started studying for an examination. Мы продолжали работать над докладом - We kept working on our research.
Modality Они могут бегать марафоны - They can run marathons. Она умеет играть на рояле - She can play piano. Он любит летать на самолете - He likes flying. Я хочу участвовать в конкурсе - I want to participate in a contest. Сестра ненавидит готовить - My sister hates cooking. Дети привыкли жить самостоятельно - Children are get accustomed to live on their own.
Pay attention how the auxiliary parts are changed in these examples.