Despite of the low transition temperature, the recently identified superconductor ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ has garnered significant interest due to its unique symmetry and magnetic characteristics, particularly the existence of two superconducting phases under a magnetic field, one of which exceeds the Pauli-Clogston limit. The field-induced transition from a low-field even-parity superconducting state to a high-field odd-parity state is usually described as a singlet-triplet transition. However, it is uncommon for a single compound to exhibit both triplet and singlet superconducting scenarios. The aim of this paper is to investigate the possibilities of symmetry changes in the superconducting state without a change of spin multiplicity. To this end, we construct the superconducting order parameter (SOP) of ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ based on Anderson pair functions, considering the phase winding within the symmetry of the point group $D_{4h}$ and the magnetic group $4/m{m}^{\prime }{m}^{\prime }$. It was found that two triplets with opposite-spin and equal-spin pairing states of symmetry $E_{1u}^{\prime +}$, are nodeless but exhibit distinct internal structures and may be associated with low- and high-field phases. Additionally, nontrivial Cooper pairing resulting from the nonsymmorphic structure of the space group was examined, particularly in the case where the Fermi surface intersects with the boundaries of a BZ (Brillouin zone). It was determined that at the $\mathrm{X}$ point, triplet pairs are even, while singlet pairs can be either even or odd. Furthermore, at the $\mathrm{X}$ point, pair density waves that alter phase by $\pi$ at the atomic centers linked by lattice translations are also feasible. To explore the possibility of such scenarios, precise DFT calculations of the band structure were performed, revealing the contribution of Ce $4f$ electrons to the states at the Fermi level. Thus the even-odd transition can take place in a triplet scenario at symmetry points of a BZ.

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• Order parameters
• Pair density wave
• Pairing mechanisms
• Superconducting order parameter
• Topological superconductors
• Topological materials
• First-principles calculations
• Methods in superconductivity