## 3 Results and Discussion  
The XRD investigation revealed three distinct structural phases in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) series within the compositional ranges studied (Figures 1 and S1, Table 1). Depending on the chemical composition, the identified phases include the following:  
1. A rhombohedral phase with lattice parameters \(a=b\approx\sqrt{2a_{\rm p}}\) and \(c\approx 2\sqrt{3a_{\rm p}}\), with reflection conditions corresponding to the space group \(R3c\)- a phase isostructural to the parent BiFeO\({}_{3}\)[39];
2. An orthorhombic phase with lattice parameters \(a\approx\sqrt{2a_{\rm p}}\), \(b\approx 2\sqrt{2a_{\rm p}}\), and \(c\approx 4a_{\rm p}\), with reflection conditions corresponding to the space group \(Pnam\);
3. An orthorhombic phase with lattice parameters \(a\approx\sqrt{2a_{\rm p}}\), \(b\approx 2a_{\rm p}\), and \(c\approx\sqrt{2a_{\rm p}}\), with reflection conditions corresponding to the space group \(Pnam\). Figure 1: Examples of observed (red circles) and calculated (black line) X-ray diffraction patterns for Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) samples, shown on a logarithmic scale along the vertical axis to highlight the low-intensity peaks: (**a**) \(y=0.15\), \(x=0.35\); (**b**) \(y=0.1\), \(x=0.45\); and (**c**) \(y=0.175\), \(x=0.35\). The magnified patterns justify the selection of the present perovskite phases (polar, \(R3c\); antipolar, \(Pnam\); and nonpolar, \(Pnam\), with the allowed Bragg reflections indicated by ticks) and exclude the presence of impurity phases. The low-intensity peaks around \(29^{\circ}\) (marked with an asterisk) result from a small \(K_{\beta}\) component of the radiation leaking through the \(K_{\beta}\) filter (a Ni foil placed before the detector), appearing as “images”, at lower angles, of the corresponding strong \(K_{\alpha}\) peaks. The intensity changes near the fundamental perovskite reflections (100)\({}_{\rm c}\) and (110)\({}_{\rm c}\) (around \(22^{\circ}\) and \(31^{\circ}\), respectively) are due to the Ni absorption edge of the \(K_{\beta}\) filter. Here, \(a_{\rm p}\approx 3.95\) A represents the parameter of the cubic perovskite subcell. It is found that the combined chemical substitution significantly increases the Ca and Mn concentrations, which define the stability ranges of the parent polar rhombohedral structure (space group \(R3c\)) [39] in the solid solutions of Bi\({}_{1-y}\)Ca\({}_{y}\)FeO\({}_{3-y/2}\) and BiFe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) (this structure exists in a single-phase state at \(y\leq 0.1\)[17; 40] and \(x<0.25\)[41], respectively). The diffraction experiment indicates that, at lower concentrations (\(x<y\)), Mn contributes to the stabilization of the polar rhombohedral phase in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) systems. This effect is primarily due to Mn doping-induced elimination of oxygen vacancies (as discussed in detail in [34]), similar to what occurs in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-y}\)Mn\({}_{y}\)O\({}_{3}\) system, where simultaneous Bi\({}^{3+}\) to Ca\({}^{2+}\) and Fe\({}^{3+}\) to Mn\({}^{4+}\) substitution extends the boundary of the ferroelectric phase to \(y=0.19\)[37]. At higher concentrations (\(x>y\)), depending on the Ca content, Mn provokes transitions either to the antipolar phase (space group \(Pnam\)) specific to BiFe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) perovskites at \(0.25<x<0.65\)[41; 42] or to the nonpolar phase (space group \(Pnam\)) characteristic of CaMnO\({}_{3}\)[43]. The concentration boundary of the polar to antipolar (nonpolar) phase transition shifts toward smaller \(x\) with an increase in \(y\) (Table 1). This observation is in line with the previous study of the Bi\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) series indicating that the polar \(R3c\) phase vanishes at \(0.1<x<0.2\)[44].