Symmetric and antisymmetric terms have been obtained in the framework of the variational approach for two-dimensional (2𝐷) Coulomb systems of symmetric trions 𝑋𝑋𝑌 . Stability diagrams and certain anomalies arising in the 2𝐷 space are explained qualitatively in the framework of the Born-Oppenheimer adiabatic approximation. The asymptotics of energy terms at large distances obtained for an arbitrary space dimensionality are analyzed, and some approximation formulas for 2𝐷 terms are proposed. An anomalous dependence of multipole moments on the space dimensionality has been found in the case of a spherically symmetric field. The main results obtained for the 2𝐷 and 3𝐷 problems of two Coulomb centers are compared.

For a two-dimensional Coulomb system of three charged particles, among which two particles are identical and the third particle is of different nature, we study the conditions of existence for symmetric and antisymmetric bound energy states (conditions of stability) in the masscharge (m, Z) plane. High-precision three-body numerical calculations based on a stochastic variational method with Gaussian bases are performed. Several anomalous effects in the behavior of the characteristic distances between particles are revealed, and the nonzero quadrupole moment is found in a two-dimensional polar-symmetric field. The systematic comparison of the results for two-and three-dimensional systems is performed. The values of energy and size, the density distributions, and the correlation functions for the various reference three-particle systems are obtained. K e y w o r d s: three charged particles, two-dimensional systems, stability, variational method, structural functions