  In this paper we consider the problem on estimates for Mittag-Leffler
functions with the smooth phase functions of two variables having singularities
of type $D_{\infty} $, $D_{4}^{\pm}$ and $A_{r}$. The generalisation is that we
replace the exponential function with the Mittag-Leffler-type function, to
study oscillatory type integrals. We extend results of paper \cite{Ruzhansky}
and \cite{Ruzhansky2021} to two-dimensional integrals with phase having some
simple singularities.
