We discuss the steady states of the alphaomega-dynamo in a thin disc w
hich arise due to alpha-quenching. Two asymptotic regimes are consider
ed, one for the dynamo number D near the generation threshold D0, and
the other for Absolute value of D >> 1. Asymptotic solutions for \D-D0
\ << \D0\ have a rather universal character provided only that the bif
urcation is supercritical. For Absolute value of D >> 1 the asymptotic
solution crucially depends on whether or not the mean helicity alpha,
as a function of B, has a positive root (here B is the mean magnetic
field). When such a root exists, the field value in the major portion
of the disc is O(1), while near the disc surface thin boundary layers
appear where the field rapidly decreases to zero (if the disc is surro
unded by vacuum). Otherwise, when alpha = O(Absolute value of B -s) fo
r Absolute value of B --> infinity, we demonstrate that Absolute value
of B = O(Absolute value of D 1/s) and the solution is free of boundar
y layers. The results obtained here admit direct comparison with obser
vations of magnetic fields in spiral galaxies, so that an appropriate
model of nonlinear galactic dynamos hopefully could be specified.