Fan triangulations of a hyperbolic plane of positive curvature

We study the families $(\mathscr F_\lambda)$ of normal partitions of a $3$-$(1)$-contour $F$ of a hyperbolic plane $\widehat H$ of positive curvature into simple $4$-contours whose hyperbolic diagonal lines are parallel to the base of $F$. A $3$-$(1)$-contour with a given partition from a family $(\mathscr F_\lambda)$ (or some its normal subpartition) is called a fan. We construct fan partitions $\mathscr P_\text e$, $\mathscr P_\text h$ and $\mathscr P_\text p$ of $\widehat H$ whose symmetry groups are generated by a shift along an elliptic (respectively, hyperbolic and parabolic) straight line. It is proved that the partitions $\mathscr P_\text h$ and $\mathscr P_\text p$ are normal. The partitions $\mathscr P_\text h$ и $\mathscr P_\text p$ whose cells are trihedrals present examples of the first triangulations of $\widehat H$.