  The $SU(2)_A \times U(2)_V$-symmetric chiral linear sigma model in the
presence of the axial anomaly is studied in the local-potential approximation
of the Functional Renormalization Group (FRG). The renormalization group (RG)
flow is investigated in a truncation which reproduces recent results for the
$U(2)_A \times U(2)_V$-symmetric model in the limit of vanishing axial anomaly
strength. We search for the conjectured O(4) fixed point in the presence of the
$U(1)_A$ anomaly and analyze its stability properties.
