  The model under consideration is the two-dimensional (2D) one-component
plasma of pointlike charged particles in a uniform neutralizing background,
interacting through the logarithmic Coulomb interaction. Classical equilibrium
statistical mechanics is studied by non-traditional means. The question of the
potential integrability (exact solvability) of the plasma is investigated,
first at arbitrary coupling constant \Gamma via an equivalent 2D
Euclidean-field theory, and then at the specific values of \Gamma=2*integer via
an equivalent 1D fermionic model. The answer to the question in the title is
that there is strong evidence for the model being not exactly solvable at
arbitrary \Gamma but becoming exactly solvable at \Gamma=2*integer. As a
by-product of the developed formalism, the gauge invariance of the plasma is
proven at the free-fermion point \Gamma=2; the related mathematical peculiarity
is the exact inversion of a class of infinite-dimensional matrices.
