  The ground-state of two-dimensional (2D) systems of classical particles
interacting pairwisely by the generalized Lennard-Jones potential is studied.
Taking the surface area per particle $A$ as a free parameter and restricting
oneself to periodic Bravais lattices with one particle per unit cell,
B\'etermin L [2018 Nonlinearity {\bf 31} 3973] proved that the hexagonal,
rhombic, square and rectangular structures minimize successively the
interaction energy per particle as $A$ increases. We show here that the
second-order transitions between the rhombic/square and square/rectangular
phases are of mean-field type. The aim of this paper is to extend B\'etermin's
analysis to periodic 2D lattices with more than one particle per elementary
cell. Being motivated by previous works dealing with other kinds of models, we
propose as the ground-state the extensions of the 2D rectangular (1-chain)
lattice, namely the "zig-zag" (2-chain), 3-chain, 4-chain, etc. structures
possessing 2, 3, 4, etc. particles per unit cell, respectively. By using a
recent technique of lattice summation we find for the standard Lennard-Jones
potential that their ground-state energy per particle approaches systematically
as the number of particles per unit cell increases to the one of a phase
separated state (the optimal hexagonal lattice). We analyze analytically the
low-density limit $A\to\infty$ and the limiting hard-core case of the
generalized Lennard-Jones potential.
