  We prove a complexity dichotomy for a class of counting problems expressible
as bipartite 3-regular Holant problems. For every problem of the form
$\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued
ternary symmetric constraint function on Boolean variables, we prove that it is
either P-time computable or #P-hard, depending on an explicit criterion of $f$.
The constraint function can take both positive and negative values, allowing
for cancellations. The dichotomy extends easily to rational valued functions of
the same type. In addition, we discover a new phenomenon: there is a set
$\mathcal{F}$ with the property that for every $f \in \mathcal{F}$ the problem
$\operatorname{Holant}\left(f\mid =_3 \right)$ is planar P-time computable but
#P-hard in general, yet its planar tractability is by a combination of a
holographic transformation by $\left[\begin{smallmatrix} 1 & 1 \\ 1 & -1
\end{smallmatrix}\right]$ to FKT together with an independent global argument.
