  We prove that given a Herglotz vector field on the unit ball of
$\mathbb{C}^n$ of the form $H(z,t)=(a_1 z_1,...,a_n z_n)+O(|z|^2)$ with $\Re
a_j<0$ for all $j$, its evolution family admits an associated Loewner chain,
which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE
admits a solution defined for all positive times.
