  A Walecki tournament is any tournament that can be formed by choosing an
orientation for each of the Hamilton cycles in the Walecki decomposition of a
complete graph on an odd number of vertices. In this paper, we show that if
some arc in a Walecki tournament on at least $7$ vertices lies in exactly one
directed triangle, then there is a vertex of the tournament (the vertex
typically labelled $*$ in the decomposition) that is fixed under every
automorphism of the tournament. Furthermore, any isomorphism between such
Walecki tournaments maps the vertex labelled $*$ in one to the vertex labelled
$*$ in the other.
  We also show that among Walecki tournaments with a signature of even length
$2k$, of the $2^{2k}$ possible signatures, at least $2^k$ produce tournaments
that have an arc that lies in a unique directed triangle (and therefore to
which our result applies).
