We study some weight-homogeneous systems which are not algebraically completely integrable (ACI) in the sense of Adler and van Moerebeke, but whose invariant level surface completes into a semi-abelian variety by adding a set of points (thus ACI in the sense of Mumford).
AMS 2000 Mathematics subject classification: Primary 37J35. Secondary 14H70; 37N05; 70E40