  The parameters 2-(36,15,6) are the smallest parameters of symmetric designs
for which a complete classification up to isomorphism is yet unknown.
Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that admit an
automorphism of odd prime order, and gave a partial classification of such
designs that admit an automorphism of order 2. In this paper, we give the
classification of all symmetric 2-$(36,15,6)$ designs that admit an
automorphism of order two. It is shown that there are exactly $1 547 701$
nonisomorphic such designs, $135 779$ of which are self-dual designs. The
ternary linear codes spanned by the incidence matrices of these designs are
computed. Among these codes, there are near-extremal self-dual codes with
previously unknown weight distributions.
