There was no wind, and except for the flowing stream, all lay in stillness, in the midst of which that single leaf alone kept up its busy fluttering, on and on.
I don't see why 'its' is there. Without 'its', this sentence seems to be grammatical. "That single leaf alone kept up busy fluttering, on and on."
anonymous that single leaf alone kept up [its busy fluttering] that single leaf alone kept up the busy fluttering that it (the leaf) was doing It (the leaf) was fluttering (busily) is expressed as a gerund clause (in brackets above). That clause is the complement of the phrasal verb kept up . It is normal to express the subject of a gerund clause in the possessive.
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anonymousthat single leaf alone kept up [its busy fluttering]
that single leaf alone kept up the busy fluttering that it (the leaf) was doing
It (the leaf) was fluttering (busily) is expressed as a gerund clause (in brackets above). That clause is the complement of the phrasal verb kept up. It is normal to express the subject of a gerun