  The nonlocal symmetry is derived from the known Darboux transformation (DT)
of the Hirota-Satsuma coupled KdV (HS-cKdV) system, and infinitely many
nonlocal symmetries are obtained by introducing some internal parameters. By
extending the HS-cKdV system to an auxiliary system with five dependent
variables, the prolongation is found to localize the nonlocal symmetry related
to the DT. Base on the enlarged system, the finite symmetry transformations and
similarity reductions about the local symmetries are computed, which lead to
some novel exact solutions of the HS-cKdV system. These solutions contain some
new solutions from old ones by the finite symmetry transformation and exact
interaction solutions among solitons and other complicated waves including
periodic cnoidal waves and Painlev\'{e} waves through similarity reductions.
Some integrable models from the nonlocal symmetry related to the DT are
obtained in two aspects: the negative HS-cKdV hierarchy obtained by introducing
the internal parameter and integrable models both in lower and higher
dimensions given by restricting the symmetry constraints.
