  We study an electronic structure of CeRhSb$_{1-x}$Sn$_x$ system, which
displays quantum critical transition from a Kondo insulator to a non-Fermi
liquid at $x=0.13$. We provide ultraviolet photoelectron spectra of valence
band obtained at 12.5 K. Acoherent peak at the Fermi level is not present in
the data, but a signal related to 4f$^1$$_{7/2}$ final state is detected.
Spectral intensity at the Fermi edge has a general tendency to grow with Sn
content. Theoretical calculations of band structure are realized with
full-potential local-orbital minimum-basis code using scalar relativistic and
full relativistic approach. The calculations reveal a depletion of density of
states at the Fermi level for CeRhSb. This gap is shifted above the Fermi
energy with increasing Sn content and thus a rise of density of states at the
Fermi level is reflected in the calculations. It agrees with metallic
properties of compounds with larger $x$. The calculations also yield another
important effects of Sn substitution. Band structure is displaced in a
direction corresponding to hole doping, although with deviations from a rigid
band shift scenario. Lifshitz transitions modify a topology of the Fermi
surface a few times and a number of bands crossing the Fermi level increases.
