  In a dynamical lattice simulation with the overlap Dirac operator and $N_f=2$
mass degenerate quarks we study all possible $J=0$ and $J=1$ correlators upon
exclusion of the low lying "quasi-zero" modes from the valence quark
propagators. After subtraction of a small amount of such Dirac eigenmodes all
disconnected contributions vanish and all possible point-to-point $J=0$
correlators with different quantum numbers become identical, signaling a
restoration of the $SU(2)_L \times SU(2)_R \times U(1)_A$. The original ground
state of the $\pi$ meson does not survive this truncation, however. In
contrast, in the $I=0$ and $I=1$ channels for the $J=1$ correlators the ground
states have a very clean exponential decay. All possible chiral multiplets for
the $J=1$ mesons become degenerate, indicating a restoration of an $SU(4)$
symmetry of the dynamical QCD-like string.
