On distance-regular graphs of diameter $3$ with eigenvalue $\theta= 1$

For a distance-regular graph $\Gamma$ of diameter $3$, the graph $\Gamma_i$ can be strongly regular for $i=2$ or $3$. J. Kulen and co-authors found the parameters of a strongly regular graph $\Gamma_2$ given the intersection array of the graph $\Gamma$ (independently, the parameters were found by A.A. Makhnev and D.V. Paduchikh). In this case, $\Gamma$ has an eigenvalue $a_2-c_3$. In this paper, we study graphs $\Gamma$ with strongly regular graph $\Gamma_2$ and eigenvalue $\theta=1$. In particular, we prove that, for a $Q$-polynomial graph from a series of graphs with intersection arrays $\{2c_3+a_1+1,2c_3,c_3+a_1-c_2;1,c_2,c_3\}$, the equality $c_3=4 (t^2+t)/(4t+4-c_2^2)$ holds. Moreover, for $t\le 100000$, there is a unique feasible intersection array $\{9,6,3;1,2,3\}$ corresponding to the Hamming (or Doob) graph $H(3,4)$. In addition, we found parametrizations of intersection arrays of graphs with $\theta_2=1$ and $\theta_3=a_2-c_3$.