The parallel between Yang-Mills theory and General Relativity is an attractive topic that motivated a lot of interesting researches. Various versions of the gauge theory of gravity were proposed which provided a deep understanding of such basis concepts as the geometry of spacetime, the connection between symmetry and dynamics. In this work we consider the problem of interaction between intrinsic degrees of freedom of particles with the Lorentz gauge field solution that has the same singular feature as for the Schwarzschild solution of General Relativity. The effective potential for a particle in the Schwarzschild-like Lorentz gauge field was investigated in comparison with Einstein and Newtonian theories of gravity. The equations of motion for particles allows planar motions that can be solved numerically. We have shown that there exit the Keplerlike orbits which correspond to certain sets of parameters and initial conditions. The obtained results can be regarded as an experimental evidence for the existence of the "internal Lorentz symmetry" of point particles that is analogue to internal unitary symmetries.