  We give a complete classification of (co)torsion pairs in finite
$2$-Calabi-Yau triangulated categories with maximal rigid objects which are not
cluster tilting. These finite $2$-Calabi-Yau triangulated categories are
divided into two main classes: one denoted by $\mathcal{A}_{n,t}$ called of
type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the
geometric model of cluster categories of type $A, $ or type $D$, we give a
geometric description of (co)torsion pairs in $\mathcal{A}_{n,t}$ or $D_{n,t}$
respectively, via defining the periodic Ptolemy diagrams. This allows to count
the number of (co)torsion pairs in these categories. Finally, we determine the
hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated
categories with maximal rigid objects which are not cluster tilting via quivers
and relations.
