  Singularities of plane into plane mappings described by parabolic
two-component systems of quasi-liner partial differential equations of the
first order are studied. Impediments arising in the application of the original
Whitney's approach to such case are discussed. Hierarchy of singularities is
analysed by double-scaling expansion method for the simplest $2$-component
Jordan system. It is shown that flex is the lowest singularity while higher
singularities are given by $(k+1,k+2)$ curves which are of cusp type for
$k=2n+1$, $n=1,2,3,\dots$. Regularization of these singularities by deformation
of plane into plane mappings into surface $S^{2+k} (\subset \mathbb{R}^{2+k} )
$ to plane is discussed. Applicability of the proposed approach to other
parabolic type mappings is noted.
