  We continue the study of the so-called thematic factorizations of admissible
very badly approximable matrix functions. These factorizations were introduced
by V.V. Peller and N.J. Young for studying superoptimal approximation by
bounded analytic matrix functions. Even though thematic indices associated with
a thematic factorization of an admissible very badly approximable matrix
function are not uniquely determined by the function itself, R.B. Alexeev and
V.V. Peller showed that the thematic indices of any monotone non-increasing
thematic factorization of an admissible very badly approximable matrix function
are uniquely determined. In this paper, we prove the existence of monotone
non-decreasing thematic factorizations for admissible very badly approximable
matrix functions. It is also shown that the thematic indices appearing in a
monotone non-decreasing thematic factorization are not uniquely determined by
the matrix function itself. Furthermore, we show that the monotone
non-increasing thematic factorization gives rise to a great number of other
thematic factorizations.
