  Effects of the completely unknown symmetry (isovector) potential of the \D on
the total and differential \rpi in heavy-ion collisions at beam energies from
100 to 1000 MeV/A are explored within an isospin-dependent transport model
IBUU. The effects are found to be negligible at beam energies above the pion
production threshold due to the very short lifetimes of less than 2 fm/c for
$\Delta$ resonances with masses around $m_{\Delta}=1232$ MeV, leaving the
$\pi^-/\pi^+$ ratios of especially the energetic pions still a reliable probe
of the high-density behavior of nuclear symmetry energy $E_{sym}(\rho)$.
However, as the beam energy becomes deeply sub-threshold for pion production,
effects of the $\Delta$ symmetry potential becomes appreciable especially on
the \rpi of low-energy pions from the decays of low-mass $\Delta$ resonances
which have lived long enough to be affected by their mean-field potentials,
providing a useful tool to study the symmetry potential and spectroscopy of
$\Delta$ resonances in neutron-rich nuclear matter. Interestingly though, even
at the deeply sub-threshold beam energies, the differential \rpi of energetic
pions remains sensitive to the \esym at supra saturation densities with little
influence from the uncertain symmetry potential of the $\Delta$ resonance.
