The authors investigate first-order weakly hyperbolic systems of PDEs with coefficients depending only on \(t\). The system is supposed to be pseudosymmetric according to a given definition (e.g., in two space dimensions, the system with the matrix \(A=(a_{ij})\) is pseudosymmetric iff \(a_{ii}\) are real and \(a_{12} \cdot a_{21} >0)\). Well-possedness of the Cauchy problem in \(H^\infty\) resp. in Gevrey classes is proved for the case of analytic resp. \(C^\infty\) coefficients.