  Ruthenium oxide with the rutile structure is one of example of altermagnets.
These systems are characterized by compensated magnetic moments (typical for
antiferromagnets) and strong time reversal symmetry breaking (typical for
ferromagnets). However, in such cases, the electronic band structure exhibit
strong spin splitting along some directions in the momentum space. Occurrence
of the compensated magnetic textures allows for realization of surfaces with
specific magnetization, which dependent on the surface orientation and/or its
termination. Here, we study interplay between the electronic surface states and
the surface magnetization. We show that the spin-resolved spectra strongly
depends on a direction in reciprocal space. Such properties can be used for the
experimental confirmation of the altermagnetism in RuO$_{2}$ within the
spectroscopic techniques. Additionally, we show that the most modified orbitals
in the system are $d_{z^{2}}$ and $d_{xy}$ orbitals of Ru. Similarly, the Ru
$e_{g}$ states are most sensitive on epitaxial strain, what can suggest some
link between altermagnetism and strain.
