The first-order differential Lévy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of (1 + d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural Z 2 -graded Lie symmetry.In this paper we show that, for a certain class of supersymmetric PDE's, a natural Z 2 ×Z 2graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the (1 + 1)-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a Z 2 × Z 2 -graded Lie superalgebra, realized by first and secondorder differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the Z 2 -and Z 2 × Z 2 -graded extensions are no longer allowed.The construction of the Z 2 × Z 2 -graded Lie symmetry of the (1 + 2)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra. The fact that a Lie superalgebra appears even in a purely bosonic setting is not so surprising. Indeed, for the harmonic oscillator, the old results of [20] can be expressed, in modern language, by stating that the Fock vacuum of creation/annihilation operators can be replaced by a lowest weight representation of an osp(1|2) spectrum-generating superalgebra.

Background aimsTransplantation of synovial mesenchymal stromal cells (MSCs) may induce repair of cartilage defects. We transplanted synovial MSCs into cartilage defects using a simple method and investigated its usefulness and repair process in a pig model.MethodsThe chondrogenic potential of the porcine MSCs was compared in vitro. Cartilage defects were created in both knees of seven pigs, and divided into MSCs treated and non-treated control knees. Synovial MSCs were injected into the defect, and the knee was kept immobilized for 10 min before wound closure. To visualize the actual delivery and adhesion of the cells, fluorescence-labeled synovial MSCs from transgenic green fluorescent protein (GFP) pig were injected into the defect in a subgroup of two pigs. In these two animals, the wounds were closed before MSCs were injected and observed for 10 min under arthroscopic control. The defects were analyzed sequentially arthroscopically, histologically and by magnetic resonance imaging (MRI) for 3 months.ResultsSynovial MSCs had a higher chondrogenic potential in vitro than the other MSCs examined. Arthroscopic observations showed adhesion of synovial MSCs and membrane formation on the cartilage defects before cartilage repair. Quantification analyses for arthroscopy, histology and MRI revealed a better outcome in the MSC-treated knees than in the non-treated control knees.ConclusionsLeaving a synovial MSC suspension in cartilage defects for 10 min made it possible for cells to adhere in the defect in a porcine cartilage defect model. The cartilage defect was first covered with membrane, then the cartilage matrix emerged after transplantation of synovial MSCs.

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