The solid solutions of Pb1-xSnxTe with x > 0.35 (at x = 0.35 the band gap Eg equals zero at temperature T = 0) behave like crystalline topological insulators (CTI). This conforms to theory and experiments, including angle-resolved photoemission spectroscopy (ARPES) data [1-3]. Studies of CTI surface conductivity conditioned by spin-polarized Dirac-cone-like surface states are of particular interest. However, the high bulk conductivity of the samples (inherent for CTI and for undoped PbSnTe) brings on quite serious complications into such studies.  In the region of 0.22 < x < 0.28 the introduction of 0.5 to 3 at.% of indium into Pb1-xSnxTe results in pinning of Fermi level in the forbidden gap, and the specific resistance ρ at T ~ 4.2 K approaches that of the conventional insulators [4]. At the surface of such insulating layers of PbSnTe:In thin films of PbSnTe with composition of x > 0.35 can be formed as appropriate. For example, molecular beam epitaxy (MBE) can proceed in such a way that the surface of PbSnTe would enrich with tin (i.e. increase of x) [5]. This also conforms to the typical for CTI results of ARPES, which the authors observed at atomically clean surface of Pb0.72Sn0.28Te:In (111)-(1x1) grown using MBE on BaF2 substrate (see Fig.).  It has been shown that the conductivity of the insulating films of PbSnTe:In due to injection from contacts and in the space charge limited current (SCLC) regime also revealed some specific features conditioned by the surface. And it is important to distinguish those related to the surface CTI conductivity from all the rest. In this work the authors present the results of studies and analyze probable reasons for such a drastic change of conductivity of MBE-grown PbSnTe:In films in SCLC regime after a chemical treatment of the surface, namely as follows: the transformations of current-voltage characteristics with the change of current by 104- 106 times, the field effect, and the angular dependences of the magnetoresistance [6].