  We study the structure of the codifferent and of additively indecomposable
integers in families of totally real cubic fields. We prove that for cubic
orders in these fields, the minimal trace of indecomposable integers multiplied
by totally positive elements of the codifferent can be arbitrarily large. This
is very surprising, as in the so far studied examples of quadratic and simplest
cubic fields, this minimum is 1 and 2. We further give sharp upper bounds on
the norms of indecomposable integers in our families.
