  We propose a new powerful family of tests of univariate normality. These
tests are based on an initial value problem in the space of characteristic
functions originating from the fixed point property of the normal distribution
in the zero bias transform. Limit distributions of the test statistics are
provided under the null hypothesis, as well as under contiguous and fixed
alternatives. Using the covariance structure of the limiting Gaussian process
from the null distribution, we derive explicit formulas for the first four
cumulants of the limiting random element and apply the results by fitting a
distribution from the Pearson system. A comparative Monte Carlo power study
shows that the new tests are serious competitors to the strongest well
established tests.
