To increase the resulting C polarization we need to consider the processes underlying the redistribution of polarization at ultra-low magnetic fields. PHIP transfer is effective only in a relatively narrow range of ultra-low fields within the region of avoided crossings of the energy levels (Level Anti-Crossing, LAC). LACs occur when two or more energy levels tend to cross, but off-diagonal terms of the Hamiltonian prevent them from crossing. The key point here is that the states involved in LAC become mixed, which enables a coherent exchange of their populations. The concept of LAC facilitates theoretical treatment of a broad variety of NMR experiments, including PHIP, SABRE, DNP and many others . In our case, we focus on LACs to describe the conversion of the initial spin order of pH2 protons, i.e., of the singlet state, into the net magnetization of the C nucleus of the target molecule. The simplest case to demonstrate LAC at ULF is a three-spin system of two protons and a C nucleus, which corresponds for example to C-maleate and C-fumarate . Assuming that only the proton singlet state is populated initially and that the sweep goes from zero to positive field values, we have to account for only one LAC at ULF (Fig. .a), between the 𝑆 0 ′ 𝛼 and 𝑇 + ′ 𝛽 states. Here, we consider the STZ' basis to describe spin states of the system. This basis is a direct product of the Zeeman basis for the C nucleus and singlet-triplet basis for two protons, taking into account the mixing of 𝑆 0 and 𝑇 0 states induced by the heteronuclear JCH coupling . The LAC depends on the JHH and JCH spin-spin coupling constants. For more complex spin systems, the detailed treatment of the LACs becomes rather complicated, unless some assumptions or simplifications are made.