Are super-symmetrical complements of physically realised theories (scalar, tensor, Yang-Mills etc.) compellingly first-order differential equations?

Physics Asked on May 9, 2021

I made the observation that all super-symmetrical extensions of "standard" theories (SM and GR) are given by first order differential equations. In particular if in case of EFEs the super-symmetrical complement is looked for, a second-order differential equation does seem out of discussion. Why is that the case? Because all super symmetrical extensions are realised by spinor fields that have to follow kind of the paradigma of the Dirac-equation? Or super-symmetry cannot be achieved with second-order differential equation complements?

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