  Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface
$M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer
and the orbit of $f$ with respect to the right action of the group of
diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author
described the homotopy types of connected components of $\mathcal{S}(f)$ and
$\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or
$\chi(M)<0$. Moreover, in two recent papers the authors considered special
classes of smooth functions on $2$-torus $T^2$ and shown that the computations
of $\pi_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk
and cylinder.
  In the present paper we consider another class of Morse functions
$f:T^2\to\mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for
every such function there exists a subsurface $Q\subset T^2$, diffeomorphic
with a cylinder, such that $\pi_1\mathcal{O}(f)$ is expressed via the
fundamental group $\pi_1\mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$.
  This result holds for a larger class of smooth functions $f:T^2\to
\mathbb{R}$ having the following property: for every critical point $z$ of $f$
the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial
$\mathbb{R}^2\to \mathbb{R}$ without multiple factors.
