The main result of the paper is the following: let (M,g) be a space-time with metric \[ (1)\quad ds^ 2=A dx^ 2_ 1-dx^ 2_ 2+B dx^ 2_ 3+C dx^ 2_ 4 \] where \(A=-a^ 2\cosh^ 2(\sqrt{k}x_ 2)\), \(B=\epsilon \cdot b^ 2\cosh^ 2(\sqrt{k}x_ 4)\), \(C=-\epsilon\), \(k\in {\mathbb{R}}\setminus \{0\}\), \(\epsilon \in \{-1,1\},a=a(x_ 1)\), \(b=b(x_ 3)\). Then (M,g) is an Einstein space. This is a generalization of the case \(a=b=1\), due to A. Z. Petrov [New methods in general relativity theory (Moscow 1966; Zbl 0146.239)]. The authors also show that there exist Einstein space-times which are non Ricci flat, have metric of type (1) with A, B arbitrary and which are of type \(V_{464}\). (The type \(V_{abc}\) was defined by the second author [Period. Math. Hung. 9, 49-53 (1978; Zbl 0386.53012)] and is given by some algebraic ranks a, b, c of the curvature tensor.)