twisted SUSY (\[defsusytr\]). However, it seems that the twisted SUSY is not enough to guarantee renormalizability.

It is obvious that different deformations obtained from different twists lead to models with different quantum properties. In our previous work [@D-def-us] we studied a deformation which preserves the full undeformed SUSY. There, after relaxing the condition of minimality of deformation, we obtained a renormalizable Wess-Zumino model. In this paper we work with a deformation given in terms of the non-SUSY-covariant derivatives. The Leibniz rule for the SUSY transformation (\[defsusytr\]) changes and the deformed action (\[Sincompfields\]) though invariant under twisted SUSY transformations, is not invariant under the undeformed SUSY transformations. For example, the term $K^{mn}K_{mn}F^3$ breaks the undeformed SUSY. On the other hand, the twisted SUSY allows this term as a part of the invariant term $P_2(\Phi\star P_2(\Phi\star \Phi))\Big|_{\theta\theta}$, see Equations (5.13) and (5.14) in [@miSUSY].

The classical properties of theories with twisted symmetries are not fully understood [@NCbookMi], [@TwistedNoeth]. For example, one cannot apply standard methods to find conserved charges and the modification of Noether theorem in the case of twisted symmetry has not been formulated yet. In this paper we analyze quantum properties of the theory with the twisted SUSY. This is 