  In this paper a complete description of the linear maps
$\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when
$n=2$ and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the
subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it
has been shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a
matrix with a certain structure. It was also shown that such preservers do not
change the nature of the Lorentz eigenvalues (that is, the fact that they are
associated with Lorentz eigenvectors in the interior or on the boundary of the
Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by
Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities,
when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in
$\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in
$\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum
preservers on $W_n = M_n$ also follows from the known description of the Pareto
spectrum preservers on $M_n$.
