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In this paper we use the explicit description of the Spin–$\frac{3}{2}$ Dirac operator in real dimension $3$ appeared in (Homma, Y., The Higher Spin Dirac Operators on $3$–Dimensional Manifolds. Tokyo J. Math. 24 (2001), no. 2, 579–596.) to perform the algebraic analysis of the space of nullsolution of the system of equations given by several Rarita–Schwinger operators. We make use of the general theory provided by (Colombo, F., Sabadini, I., Sommen, F., Struppa, D. C., Analysis of Dirac systems and computational...