<p>We propose a systematic method to solve viscoelastic model for viscometric flows. The method is to generalize integration factor for tensor differential equation. If the analytical solution for viscometric flow exists, then it is easier to check the validity of the constitutive equation than use of numerical solution. Our ansatz gives the analytical solutions of quasi-linear viscoelastic models, such as the upper-convected, lower-convected, and corotational Maxwell models, because they become sets of linear ordinary differential equations for viscometric flows such as simple shear and elongational flows. We expect that the integration factor method can improve the numerical algorithms for nonlinear viscoelastic models and we are in preparation of a new numerical algorithm based on the integration factor method.</p> Graphical abstract <p></p>