  This article considers the minimal non-zero (= indecomposable) solutions of
the linear congruence $1\cdot x_1 + \cdots + (m-1)\cdot x_{m-1} \equiv 0 \pmod
m$ for unknown non-negative integers $x_1, \ldots, x_n$, and characterizes the
solutions that attain the Eggleton-Erd\H{o}s bound. Furthermore it discusses
the asymptotic behaviour of the number of indecomposable solutions. The results
have direct interpretations in terms of zero-sum sequences and invariant
theory.
