Strongly regular graphs with nonprincipal eigenvalue 4 and its extensions

J. Koolen suggested the problem of investigation distance-regular graphs with strongly regular local subgraphs having the second eigenvalue $\le t$ for some natural number $t$. Earlier the Koolen problem for $t=3$ was solved. In this paper it is developed the programm of investigation distance-regular graphs with strongly regular local subgraphs having the second eigenvalue $r,$ $3<r\le4$.