## 2 Materials and Methods  
The hexagonal perovskites Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) were synthesized via the solid-state route. The compound Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) was obtained for comparison with the new phase Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) by the method described in [56]. The preliminary dried BaCO\({}_{3}\) (99.9999% purity, Vekton, RF), In\({}_{2}\)O\({}_{3}\) (99.99% purity, Reachim, RF), Al\({}_{2}\)O\({}_{3}\) (99.99% purity, Reachim, RF) were used as the starting materials; all reactants were mixed in the stoichiometric ratio and ground in an agate mortar for 1 h.  
The obtained mixture of powders was calcined at the temperature range of 800-1200 \({}^{\circ}\)C with several intermediate grindings; the time for each heat treatment was 24 h.  
For the ceramic samples, the obtained powder was pressed into pellets and sintered at the temperature 1400 \({}^{\circ}\)C for 24 h.  
For the hydration process investigations, the hydrated samples of the Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) were obtained. Hydrated samples preparation was performed according to the sequent method: the investigated compound powder was annealed at 1100 \({}^{\circ}\)C under a flow of dry argon to remove possible adsorbed water and surface carbonates, and then cooled to 200 \({}^{\circ}\)C at the rate of 1 \({}^{\circ}\)C/min in wet argon. This lower temperature was chosen to avoid adsorption of water in the sample. The wet atmosphere for the hydrated samples preparation and further experiments was obtained by bubbling the gas (argon or air) through the potassium bromide (KBr) saturated solution (\(pH_{2}O=1.92\times 10^{-2}\) atm). Phase purity was controlled by X-ray powder diffraction analysis; the measurements were performed on an ARL EQUINOX 3000 (Thermo Fisher Scientific, Waltham, MA, USA) diffractometer at room temperature with CuK\(\alpha\) radiation at the angle range 10-90\({}^{\circ}\) with steps of 0.024\({}^{\circ}\). Cell parameter calculations were performed using FullProf software. Infrared (IR) spectroscopy analysis was used to identify hydrogen-oxygen groups in the hydrated samples. Investigations were performed using the diffuse reflection technique on a Nicolet 6700 (Thermo Fisher Scientific, USA) FT-IR spectrometer at room temperature. Thermogravimetric analysis (TG) was used to determine the proton concentration in the hydrated samples; the measurements were performed on a Pyris 1 (PerkinElmer, Waltham, MA, USA) TG analyzer. At the first step, the powder sample was heated at 1000 \({}^{\circ}\)C under a flow of dry argon with heating rate of 1 \({}^{\circ}\)C/min, after that the sample was cooled to 25 \({}^{\circ}\)C with a cooling rate of 1 \({}^{\circ}\)C/min in the wet argon atmosphere. Data collection was performed during sample cooling in order to detect the increase of mass during hydration and the temperature range at which it takes place. Transport properties were investigated using Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) ceramic samples. The polished surfaces of the sintered pellets were coated with palladium-silver paste, and then fired by electrodes at 900 \({}^{\circ}\)C for 3 h. The _ac_ conductivity of the samples was measured by the 2-probe impedance spectroscopy technique using a Z-3000X (Elins, Chernogolovka, Russian Federation) frequency response analyzer over the frequency range of 100 Hz-3 MHz. The measurements were carried out at the temperature range of 300-900 \({}^{\circ}\)C with a cooling rate of 1 \({}^{\circ}\)C/min in atmospheres of dry and wet air. The data were collected during cooling at temperature intervals of 20 \({}^{\circ}\)C with the time of equilibrium of 30 min. The dry atmosphere was obtained by circulating gas through a drying column containing phosphorous pentoxide powder P\({}_{2}\)O\({}_{5}\) (\(pH_{2}O=3.5\times 10^{-5}\) atm).