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A supersymmetric generalization of the Lieb-Liniger-Yang dynamics governing$N$ massive bosons moving on a line with delta interactions among them atcoinciding points is developed. The analysis of the delicate balance betweenintegrability and supersymmetry, starting from the exactly solvable nonsupersymmetric LLY system, is one of the paper main concerns. Two extremeregimes of the $N$ parameter are explored: 1) For few bosons we fall in therealm of supersymmetric quantum mechanics with a short number of degrees offreedom, e.g., the SUSY P$\ddot{\rm o}$sch-Teller potentials if $N=1$. 2) Forlarge $N$ we deal with supersymmetric extensions of many body systems in thethermodynamic limit akin, e.g., to the supersymmetric Calogero-Sutherlandsystems. Emphasis will be put in the investigation of the ground statestructure of these quantum mechanical systems enjoying ${\cal N}=2$ extendedsupersymmetry without spoiling integrability. The decision about wether or notsupersymmetry is spontaneously broken, a central question in SUSY quantummechanics determined from the ground state structure, is another goal of thepaper.

{\cal N}=2 Supersymmetric quantum mechanics of N Lieb-Liniger-Yang bosons on a line