there are a plane in the sky.介词短语in the sky是定语
还是状语？它与哪个词有语法关系？i am busy with my work.和i help tom with his english.介词短语是什么句子成分？它与哪个词有语法关系？what do farmers do with their land.同上一问。walking is good for health.句中为什么没有冠词？there are two girls looking after them.动名词短语是什么成分，与哪个词有语法关系？

1) The existential sentences
Sentences like "There is a plane in the sky" and "there are two girls looking after them" are called existential sentences, asserting the existence or non-existence of somebody or something (existence is a rather complicated issue in the phylosophy of language).The widely-accepted analysis in the generative theory assumes that the derivation starts from a small clause which is a lexical projection in contrast to a functional projection.

(a) [a plane in the sky]

Then it is merged with the copula "be" and the pleonastic word "there" to yield the sentence "there is a plane in the sky". In this analysis, the small clause assumption stipulates that phrases like "in the sky" and "looking after them" are predicateds.

In conventional grammar, these phrases are regarded as post-modifiers. Please see Quirk et al for some useful discussion.

2)In an answer given the other day, it waspointed out that WITH as a preposition has three major uses: a) to indicate the commutative case like "I played with John". b) to mark the instrumental case such as "I cut the meat with a knife". c) to form a relational phrase, and all the examples you have produced fall into this category. These phrases are selected by the verb (or the predicate) in each sentence, and they function as the complement of the verb. Roughly speaking, WITH in these cases can be paraphrased by the phrase "in relation to".

3)Definite vs.generic information
The proposition conveyed by the sentence "walking is good for health" offers a generalization that, in turn, provides just some generic information. So no article can be used in such a case. To be more concrete, let us take a close look at the subject. WALKING is a gurend here referring to a kind of action in general not a particular event. And thus, we cannot use the indefinite article with it. In addition, when the proposition is made, it is not anchored spatially or temporally, so we cannot use the definite article with WALKING in this sentence. This analysiscan be extended to the noun HEALTH.