  We study the SYK model in the large $N$ limit beyond the replica-diagonal
approximation. First we show that there are exact replica-nondiagonal solutions
of the saddle point equations for $q=2$ for any finite replica number $M$. In
the interacting $q=4$ case we are able to construct the numerical solutions,
which are in one-to-one correspondence to the analytic solutions of the
quadratic model. These solutions are singular in the $M \to 0$ limit in both
quadratic and quartic interaction cases. The calculations of the on-shell
action at finite integer $M$ show that the nondiagonal replica-symmetric
saddles are subleading in both quadratic and quartic cases. We also study
replica-nondiagonal solutions of the SYK in the strong coupling limit. For
arbitrary $q$ we show that besides the usual solutions of the replica-diagonal
saddle point equations in the conformal limit, there are also
replica-nondiagonal solutions for any value of $M$ (including zero). The
specific configurations that we study, have factorized time and replica
dependencies. The corresponding saddle point equations are separable at strong
coupling, and can be solved using the Parisi ansatz from spin glass theory. We
construct the solutions which correspond to the replica-symmetric case and to
one-step replica symmetry breaking. We compute the regularized free energy on
these solutions in the limit of zero replicas. It is observed that there are
nondiagonal solutions with the regularized free energy lower than that of the
standard diagonal conformal solution.
