## 3 Results and Discussion
The tentative lines indicating the region of \(R3c\) and _Pnma_ phase coexistence were drawn based on the phase ratios in the phase-separated samples. Room-temperature measurements of magnetization of the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) compounds as a function of the magnetic field revealed linear or nearly linear \(M\) (\(H\)) dependencies, which do not allow for an unambiguous interpretation of the magnetic state of the samples, thus motivating further investigation at lower temperatures. These measurements found the unusual magnetization behavior correlating with the crystal structure (phase composition) of the solid solutions. The field dependencies of magnetization measured at temperatures ranging from 300 to 125 K are shown in Figures 4 and 5. The full hysteresis loops are provided in the Supplementary Materials (Figure S3). One can see that the dependencies obtained for the samples possessing the rhombohedral structure (Figure 4a,b,d,e) exhibit a change in slope indicating the onset of a metamagnetic transformation. The threshold field of the transition decreases with an increase in Mn content (i.e., with increasing \(x\)). The rhombohedral samples are characterized by very small remnant magnetization (about 0.01 emu/g at \(T\) = 125 K; see Figure 5a,b,d,e), which is far less than the spontaneous magnetization specific to the weak ferromagnetic phase of the Ca-substituted BiFeO\({}_{3}\)-based multiferroics (up to \(\sim\)0.3 emu/g at room temperature) [17; 18; 19]. Contrary to the \"\(R3c\)\" compounds, the single-phase orthorhombic samples demonstrate no features implying the existence of a metamagnetic transformation (Figure 4f,i). These samples exhibit the appearance and growth of spontaneous magnetization below room temperature (Figure 5f,i). The value of the magnetization suggests that the orthorhombic samples can be characterized as canted antiferromagnets below their Neel points. The field dependencies obtained for structurally inhomogeneous samples (Figures 4c,g,h and 5c,g,h) demonstrate features characteristic of both the rhombohedral (metamagnetic behavior) and orthorhombic (appearance of relatively large remnant magnetization) solid solutions. The remnant magnetization increases and the metamagnetic anomaly becomes less pronounced as the amount of the rhombohedral phase (Table 1) decreases. Considering the increasing complexity of the magnetic hysteresis loops as the temperature decreases, further insight into the peculiarities of the field-induced transformations in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) compounds necessitates a preliminary examination of their temperature dependencies of magnetization, \(M\) (\(T\)). The \(M\) (\(T\)) dependencies obtained for the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) samples in zero field cooling (ZFC), field-cooled cooling (FCC), and field-cooled warming (FCW) modes (Figure 6) exhibit clear regularities that clarify the composition- and temperature-driven evolution of their magnetic state. Careful inspection of the dependencies allows the identification of anomalies indicating the onset of long-range antiferromagnetic ordering (see the insets of Figure 6). In the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) (\(0.1\leq y\leq 0.175\), \(0.35\leq x\leq 0.45\)) series, the Neel temperatures (Table 1) are found to decrease at a nearly constant rate with an increase in Ca (Figure 7, left panel) and Mn (Figure 7, right panel) concentrations, with the effect of Mn substitution on the magnetic transition temperature being more pronounced. All the samples demonstrate a small low-temperature anomaly, a bifurcation between ZFC and FC dependencies, with a maximum on the ZFC curve (Figure 6), a feature typical of glassy magnetic materials. The anomaly becomes more pronounced in the samples enriched with Ca and Mn. The origin of this phenomenon has been thoroughly investigated in our previous articles [33; 34].