  We study low-energy properties of the Anderson impurity under a finite bias
voltage $V$ using the perturbation theory in $U$ of Yamada and Yosida in the
nonequilibrium Keldysh diagrammatic formalism, and obtain the Ward identities
for the derivative of the self-energy with respect to $V$. The self-energy is
calculated exactly up to terms of order $\omega^2$, $T^2$ and $V^2$, and the
coefficients are defined with respect to the equilibrium ground state. From
these results, the nonlinear response of the current through the impurity has
been deduced up to order $V^3$.
