  A supersymmetric generalization of the Lieb-Liniger-Yang dynamics governing
$N$ massive bosons moving on a line with delta interactions among them at
coinciding points is developed. The analysis of the delicate balance between
integrability and supersymmetry, starting from the exactly solvable non
supersymmetric LLY system, is one of the paper main concerns. Two extreme
regimes of the $N$ parameter are explored: 1) For few bosons we fall in the
realm of supersymmetric quantum mechanics with a short number of degrees of
freedom, e.g., the SUSY P$\ddot{\rm o}$sch-Teller potentials if $N=1$. 2) For
large $N$ we deal with supersymmetric extensions of many body systems in the
thermodynamic limit akin, e.g., to the supersymmetric Calogero-Sutherland
systems. Emphasis will be put in the investigation of the ground state
structure of these quantum mechanical systems enjoying ${\cal N}=2$ extended
supersymmetry without spoiling integrability. The decision about wether or not
supersymmetry is spontaneously broken, a central question in SUSY quantum
mechanics determined from the ground state structure, is another goal of the
paper.
