Summary: Let \((M,g)\) be a time-oriented Lorentz manifold, \(F\) a closed 2-form on \(M\), and \(u:M\to\mathbb{R}\) a smooth positive function. In this paper, the quadruple \(\Gamma= (M,g, F,u)\) is called a gyroscopic system of relativistic type. Its motions are future-directed solutions of the system of equations \[ (\nabla/ds) (dy/ds)= F^*(dy/ds) -\text{grad} u(y), \tag{1} \]
\[ g(dy/ds,dy/ds) +2u(y)=0, \tag{2} \] where \(\nabla\) is the operator of covariant differentiation on \((M,g)\), and \(F^*\) is the field of linear operators defined by \(g(F^*(X),Y) =F(X,Y)\). Let \(p\) be a point in \(M\) and \(q\) a point in the chronological future \(I^+(p)\). Let \(O_{pq}\) denote the set of piecewise smooth curves \(y:[0,\delta] \to M\) with \(\delta\in \mathbb{R}_+ =(0,\infty)\) that satisfy \[ x (0)= p, \quad x(\delta)= q. \tag{3} \] In this paper we obtain conditions on the system \(\Gamma\), the points \(p,q\), and a homotopy class \(D\in\pi_0 (O_{pq})\) that guarantee that the two-point boundary value problem (1)–(3) has solutions belonging to \(D\). Examples are considered where the gyroscopic system \(\Gamma\) describes the motion of a charged test particle in various gravitational and electromagnetic fields. In the first example, \((M,g)\) is the four-dimensional Robertson-Walker space and the field \(F\) is arbitrary. In the second example, special cases of the manifold \((M,g)\) are the external space-time of Reissner-Nordström and the spacetime of Schwarzschild-Ernst (the external space-time of the Schwarzschild black hole in the Melvin magnetic universe); a contribution to the form of the gyroscopic forces \(F\) is made by the electric and the magnetic field of the charged black hole of Reissner-Nordström, by the magnetic field of the Melvin universe, and also by an arbitrary external electromagnetic field not affecting the geometry of the space-time.