The authors consider an invertible system \[ {dX\over dt} =(L+M)X +\Phi (X),\;X \in \mathbb{R}^4, \tag{1} \] where the matrix \(L\) is a \(4\times 4\) Jordan block with a zero eigenvalue, \(M\) is a matrix of small parameters, and \(\Phi(X)\) has a Taylor series without free and linear terms. Such a system arises in hydrodynamics.
Solutions of system (1) are studied in the neighborhood of the equilibrium point \(X=0\) via the scheme of normalizing transformations. The main new result is the proof of existence of periodic and quasi-periodic solutions of system (1) in the vicinity of the resonance 1:1. In the initial problem on surface water waves, these solutions are associated with periodic and quasi-periodic waves that exist for Bond numbers \(b< {1\over 3}\), in particular, for \((b-{1\over 3})/(\widetilde \lambda- 1)^2< -{5 \over 4}\), where \(\widetilde\lambda\) is the Froude number.