Stationary Solutions of Second-Order Equations for Point Fermions in the Schwarzschild Gravitational Field

When using a second-order Schrödinger-type equation with the effective potential of the Schwarzschild field, the existence of a stationary state of half-spin particles with energy E = 0 is proved. For each of the values of quantum numbers j, l, the physically meaningful energy E = 0 (the binding energy is $E_b=m{c}^2$) is implemented at the value of the gravitational coupling constant $\alpha ⩾{\alpha }_{min}$. The particles with E = 0 are, with the overwhelming probability, at some distance from the event horizon within the range from zero to several fractions of the Compton wavelength of a fermion depending on value of the gravitational coupling constants and the values of j, l. In this paper, similar solutions of the second-order equation are announced for bound states of fermions in the Reissner–Nordström, Kerr, Kerr–Newman fields. Atomic-type systems (the point sources of the Schwarzschild gravitational field) with fermions in bound states are proposed as particles of dark matter.