  The main result of this work is to present the complete list of
Uq(sl2)-symmetries of quantum plane. For that, the structure of quantum plane
automorphisms is used. Our idea in classifying the above symmetries is in
introducing some special symbolic matrices to label the series of actions in
question. The matrices depict which generators of Uq(sl2) produce nonzero
images by acting on the generators of quantum plane. This data determines the
weights of an action unambiguously. It turns out that there are uncountably
many isomorphism classes of the symmetries. Another result is in describing
composition series of the corresponding representations. The classical limit of
the above symmetries is given.
