  Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in
modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to
\mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M
v + u, \end{equation*} where $u\neq0$. We are interested in the properties of
the attractor of the iterated function system (IFS) generated by
  $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A =
T_m(A) \cup T_p(A)$. Our two main results are as follows:
  1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in
absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior.
  2. For almost all non-degenerate IFS, the set of points which have a unique
address is of positive Hausdorff dimension -- with the exceptional cases fully
described as well.
  This paper continues our work begun in [11].
