## 1 Introduction  
The rising of the environmental challenges and the increasing in the need for safe and effective devices for energy conversion and storage mark the new decade [1; 2; 3; 4]. This makes the task of creating high-efficiency solid oxide devices for electrochemical applications very important for now. The oxygen- and proton-conducting solid oxides may be components of various devices including fuel cells and electrolyzers [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Most well-known electrolytes for medium-temperature solid oxide fuel cells (500-700 \({}^{\circ}\)C) are characterized by the perovskite or perovskite-related structure ABO\({}_{3-8}\)[22]. As an example, one of the most investigated protonic conductors, barium zirconate [23], has the structure of ordinary perovskite with a cubic unit cell (Figure 1a). The varying of the cations A and B can create the materials with a layered perovskite-related structure where the perovskite layers alternate with rock-salt layers.  
The general formula of layered perovskite-related structures can be written as AO(A\({}^{\prime}\)BO\({}_{3}\))\({}_{n}\), where \(n\) is the number of polyhedra layers stacking in the perovskite block. When \(n=1\), this formula can be represented as AA\({}^{\prime}\)BO\({}_{4}\), and the compositions with this formula are characterized by the monolayered perovskite-related structure. The family of proton-conducting oxides with this type of the structure was discovered several years ago, and it is represented by the matrix compositions SrLaInO\({}_{4}\)[24; 25; 26; 27; 28], BaLaInO\({}_{4}\)[29; 30; 31; 32; 33; 34; 35; 36; 37], BaNdInO\({}_{4}\)[38; 39; 40; 41; 42; 43], BaNdScO\({}_{4}\)[44]. The symmetry of these compositions is lower than cubic, and it is orthorhombic or monoclinic depending on the combination of cations in the A and B sublattices. As an example, the orthorhombic structure of the compound BaLaInO\({}_{4}\) is represented in Figure 1b. It was shown that various types of doping (donor and acceptor) lead to the increase in the values of ionic conductivity [45]. Moreover, the compositions based on BaLaInO\({}_{4}\) demonstrate almost pure protonic conductivity under wet air conditions at 450 \({}^{\circ}\)C [45].  
The next member of the homologous series of the Ruddlesden-Popper type of structure with \(n\) = 2 can be writing in the general case as AA\({}_{2}\)\({}^{\prime}\)B\({}_{2}\)O\(\gamma\) and as an example BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\). Despite the possibility of protonic transport in the two-layered complex oxide, BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) is not yet described; the structure of BaLa\({}_{n}\)In\({}_{n}\)O\({}_{3n+1}\) (_n_ = 1, 2) is described in detail by [PERSON] et al. [46] and [PERSON] et al. [47]. One of the main differences between BaLaInO\({}_{4}\) and BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) compositions is the ratio of perovskite blocks and rock-salt layers in the unit cell. This ratio is 1:1 for BaLaInO\({}_{4}\) and 2:1 for BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\). It can be assumed that an increase in the ratio of perovskite blocks can lead to the increase in the ionic conductivity of solid oxide. Based on this, the goal of this work is the investigation of ionic transport (O\({}^{2-}\), H\({}^{+}\)) in the two-layered compound BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) in comparison with the mono-layered compound BaLaInO\({}_{4}\). Proton transport is not previously reported for \(n\) = 2 RP phases in the Ba-La-In-O system.