  We consider a non-ideal hot pion gas with the dynamically fixed number of
particles in the model with the $\lambda\phi^4$ interaction. The effective
Lagrangian for the description of such a system is obtained after dropping the
terms responsible for the change of the total particle number. Reactions
$\pi^+\pi^-\leftrightarrow\pi^0\pi^0$, which determine the isospin balance of
the medium, are permitted. Within the self-consistent Hartree approximation we
compute the effective pion mass, thermodynamic characteristics of the system
and the variance of the particle number at temperatures above the critical
point of the induced Bose-Einstein condensation when the pion chemical
potential reaches the value of the effective pion mass. We analyze conditions
for the condensate formation in the process of thermalization of an initially
non-equilibrium pion gas. The normalized variance of the particle number
increases with a temperature decrease but remains finite in the critical point
of the Bose-Einstein condensation. This is due to the non-perturbative account
of the interaction and is in contrast to the ideal-gas case. In the kinetic
regime of the condensate formation the variance is shown to stay finite also.
