Catalyzes the conservative, sequence-specific DNA breakage and reunion reaction that generates two hairpin telomeres from a replicated telomere substrate. Breaks two phosphodiester bonds in a single DNA duplex and joins each end with the opposite DNA strand to form covalently closed hairpin telomeres (PubMed:11804598, PubMed:12753185). In vitro relaxed-circular, open-circular and linearized plasmids, but not supercoiled DNA, are all substrates (PubMed:11804598). Cleavage is position-dependent relative to conserved sequence elements (PubMed:12753185). No cofactors were found to be necessary. Found at the end of hairpin DNA molecules in the area equivalent to eukaryotic telomeres. This strain of B.burgdorferi has at least 12 linear replicons with covalently closed hairpin ends.