Nil-clean integral
![$2\times 2$](img21.png)
matrices are investigated starting with a
characterization which involves a Diophantine second degree equation. Mainly
similarity classes of matrices
A with
![$1-4\det (A)<0$](img23.png)
are determined (that
is, the elliptic case). The strongly nil-clean matrices are completely
determined and a large class of uniquely nil-clean matrices is found. In
particular, a class of uniquely nil-clean matrices which are not strongly
nil-clean is found.