Colloquial English especially is full of VP's which consist of a verb plus an adverb or preposition. Thus sentence (2) in Unit 6 might have read:
9. Se{ MCl[ He might have run away SCl[ while they were looking for him]]}
RUN AWAY is a PHRASAL VERB, ie. a main verb plus an adverb. We know that the second element is an adverb because there is no noun associated with it. LOOK FOR is a PREPOSITIONAL VERB, ie. a main verb plus the preposition for, which is associated with the pronoun him.
The obvious analyses according to our system are:
However, some people are unhappy with these analyses because they feel that the semantic unity of the verbs is disregarded. A case could be made for (might have run away) as a single unit, but this is difficult because we now have two headwords within the same phrase, which is against the rules of the grammar we are using. If you prefer this solution, then you should take run away as a single main verb:
This may seem like a long discussion of a relatively minor point. However, it illustrates an important theoretical point: no grammar as yet devised is powerful enough to cope with all the complexities and anomalies of language. There are always "grey areas" where alternative solutions are possible. If you can make out a good case for an alternative solution, then who is to say you are wrong?