  Let $\cal{P}$ be the family of all 2-connected plane triangulations with
vertices of degree three or six. Gr\"{u}nbaum and Motzkin proved (in the dual
terms) that every graph $P \in \cal{P}$ is factorable into factors $P_0$,
$P_1$, $P_2$ (indexed by elements of the cyclic group $Q = \{0,1,2\}$) such
that every factor $P_q$ consists of two induced paths with the same length
$M(q)$, and $K(q)-1$ induced cycles with the same length $2M(q)$. For $q \in
Q$, we define an integer $S^+(q)$ such that the vector $(K(q), M(q), S^+(q))$
determines the graph $P$ (if $P$ is simple) uniquely up to
orientation-preserving isomorphism. We establish arithmetic equations that will
allow calculate the vector $(K(q+1), M(q+1), S^+(q+1))$ by the vector $(K(q),
M(q), S^+(q))$, $q \in Q$. We present some applications of the equations. The
set $\{(K(q), M(q), S^+(q)): q \in Q\}$ is called the orbit of $P$. We
characterize one point orbits of graphs in $\cal{P}$. We prove that if $P$ is
of order $4n +2$, $n \in\mathbb{N}$, than it has a Hamilton bond such that the
end-trees of the bond are equitable 2-colorable and have the same order. We
prove that if $M(q)$ is odd and $K(q) \geqslant \frac{M(q)}{3}$, then there are
two disjoint induced paths of the same order, which vertices together span all
of $P$.
