  We present several examples of supersymmetric quantum mechanical systems with
weak superalgebra $su(N|1)$. One of them is the weak $su(N|1)$ oscillator. It
has a singlet ground state, $N +1$ degenerate states at the first excited
level, etc. Starting from the level $k = N+1$, the system has complete
supersymmetric multiplets at each level involving $2^N$ degenerate states. Due
to the fact that the supermultiplets are not complete for $k \leq N$, the
Witten index represents a nontrivial function of $\beta$.
  This system can be deformed with keeping the algebra intact. The index is
invariant under such deformation. The deformed system is not exactly solved,
but the invariance of the index implies that the energies of the states at the
first $N$ levels of the spectrum are not shifted, and we are dealing with a
quasi-exactly solvable system.
  Another system represents a weak generalisation of the superconformal
mechanics with $N$ complex supercharges. Also in this case, starting from a
certain energy, the spectrum involves only complete supersymmetric $2^N$-plets.
(There also exist normalizable states with lower energies, but they do not have
normalizable superpartners. To keep supersymmetry, we have to eliminate these
states.)
