  We explore ``weak'' supersymmetric systems whose algebra involves, besides
Poincare generators, extra bosonic generators not commuting with supercharges.
This allows one to have inequal number of bosonic and fermionic 1--particle
states in the spectrum. Coleman--Mandula and Haag--Lopuszanski--Sohnius
theorems forbid the presence of such extra bosonic charges in {\it interacting}
theory for $d \geq 3$. However, these theorems do not apply in one or two
dimensions. For $d=1$, we construct a nontrivial interacting system
characterized by weak supersymmetric algebra. It is related to ``n--fold''
supersymmetric systems and to quasi-exactly solvable systems studied earlier.
