  We consider Nehari's problem in the case of non-uniqueness of solution. The
solution set is then parametrized by the unit ball of $H^{\infty}$ by means of
so-called {\em regular generators} -- bounded holomorphic functions $\phi$. The
definition of {\em regularity} is given below, but let us mention now that 1)
the following assumption on modulus of $\phi$ is sufficient for {\em
regularity}: $\frac{1}{1-|\phi|^2}\in L^1(\mathbb{T})$; 2) there is no
necessary and sufficient condition of {\em regularity} on bounded holomorphic
$\phi$ in terms of $|\phi|$ on $\mathbb{T}$, \cite{Kh1}. This makes reasonable
the attempt to find a weaker sufficient condition on $|\phi|$ than the
condition in 1). This is done here. Also we are discussing certain new
necessary and sufficient conditions of {\em regularity} in terms of bounded
mean (weighted) oscillations of $\phi$. They involve the matrix $A_2$ condition
from \cite{TV}.
