## 1 Introduction  
For the last few decades, perovskite oxides have drawn great attention from researchers and have been studied intensively due to their wide application [1; 2; 3; 4; 5; 6; 7; 8; 9]. The suitable modification of the composition of oxide perovskites makes it possible to use them as electrolytes. Under appropriate conditions, they can exhibit both oxygen-ion [10; 11; 12] (under dry conditions) and proton transport [13; 14; 15] (under wet conditions and low temperatures). In the latter case, such materials have received great attention from researchers, since they can exhibit high proton conductivities at temperatures below 500 \({}^{\circ}\)C [16] compared with oxygen-ion transport reaching high values at temperatures >800 \({}^{\circ}\)C. Accordingly, intermediate temperature electrochemical devices (400-700 \({}^{\circ}\)C) for power generation technologies based on such materials will be more cost-effective compared to high-temperature ones. For example, proton conductors based on perovskites have found applications as electrolytes of solid oxide fuel cells (SOFCs). Such fuel cells, called protonic ceramic fuel cells (PCPCs or PC-SOFCs), are fundamentally different from classical SOFCs (with oxygen-ion conducting electrolyte), not only because of their lower operating temperatures, but also because they use hydrogen as fuel, and the water is generated at the cathode and does not dilute the fuel at the anode, which increases the efficiency of the device [16; 17; 18; 19; 20]. All of this stimulates the intensive search and investigation of new materials based on complex oxides, which are capable of exhibiting proton conductivity. The efforts of scientists are directed to the development of highly dense ceramic materials with the highest proton conductivity and increased chemical stability. At present, perovskites and perovskite-related structures are the most studied systems as protonic electrolytes [21; 22], although various classes of protonics are also being developed: scheelites [23; 24; 25; 26; 27; 28; 29], pyrochlores [30; 31; 32; 33; 34; 35], Ruddlesden-Popper structures [36; 37; 38], apatites [39; 40; 41], La\({}_{28-x}\)W\({}_{4+x}\)O\({}_{54+8}\)-family [42; 43; 44; 45; 46; 47; 48]. Usually, proton transport occurs via a Groththuss mechanism, which involves the fast rotation of the proton around an oxygen atom and it hopping toward a neighboring oxygen atom [49]. This mechanism is typical for structures with corner-sharing octahedral frameworks. At the same time, some features of the proton transport are described for structures with a tetrahedral environment [50; 51; 52; 53]. All this suggests the need to study and understand the regularities of proton transport for a purposeful search for new promising systems with fast proton transport. Recently, proton transport has been discovered in complex structures, for example, in the cation-deficient hexagonal perovskite-related compounds with disordered combinations of palmierite and perovskite layers [54], and later in anion-deficient hexagonal perovskite-related oxides, characterized by the intergrowth of blocks of different structures [55]. Quite recently, we have described the novel proton conductor with a hexagonal perovskite-like structure Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\)[56; 57]. Various dopants were used (Nb\({}^{5+}\), In\({}^{3+}\)) to determine the effect of the defect structure on proton transport. Although the structure of this compound was described long ago [58], the electrical properties were not studied until 2022. Therefore, it can be said that a new class of promising proton conductors can be developed based on this phase. This is demonstrated by the high proton conductivity exhibited by the doped hexagonal perovskite Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\). The crystal structure of Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) can be considered as an intergrowth structure with the alternation of a perovskite BaZrO\({}_{3}\) block and two Ba\({}_{2}\)InAlO\({}_{5}\)-blocks [58; 59].