On a vertex-symmetric graph with intersection array {205, 136, 1; 1, 68, 205}

A. Makhnev and D. Paduchikh found intersection arrays of distance-regular graphs that are locally strongly regular with the second eigenvalue 3. A. Makhnev and M. Samoilenko added to this list the intersection arrays {196, 76, 1; 1, 19, 196} and {205, 136, 1; 1, 68, 205}. However, graphs with these intersection arrays cannot be locally strongly regular. The existence of graphs with these intersection arrays is unknown. We find possible orders and fixed-point subgraphs for the elements of the automorphism group of a distance-regular graph with intersection array {205, 136, 1; 1, 68, 205}. It is proved that a vertex-transitive distance-regular graph with this intersection array is a Cayley graph.