  We present an analytical solution of an effective field theory which, in one
of its formulations, is equivalent to the Ginzburg's $\Psi$-theory for the
behavior of the Casimir force in a film of $^4$He in equilibrium with its vapor
near the superfluid transition point. We consider three versions of the theory,
depending on the way one determines its parameters from the experimental
measurements. We present exact results for the behavior of the order parameter
profiles and of the Casimir force within this theory, which is characterized by
$d=3$, $\nu=2/3$ and $\beta=1/3$, where $d$ is the bulk spatial dimension and
$\nu$ and $\beta$ are the usual critical exponents. In addition, we revisit
relevant experiments \cite{GC99} and \cite{GSGC2006} in terms of our findings.
We find reasonably good agreement between our theoretical predictions and the
experimental data. We demonstrate analytically that our calculated force is
attractive. The position of the extremum is predicted to be at $x_{\rm
min}=\pi$, with $x=(L/\xi_0)(T/T_\lambda-1)^{1/\nu}$, which value effectively
coincides with the experimental finding $x_{\rm min}=3.2\pm 0.18$. Here $L$ is
the thickness of the film, $T_\lambda$ is the bulk critical temperature and
$\xi_0$ is the correlation length amplitude of the system for temperature
$T>T_\lambda$. The theoretically predicted position of the minimum does not
depend on the one adjustable parameter, $M$, entering the theory.
