We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space \(\mathbb {R}^4_1\) which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following basic classes of timelike meridian surfaces of elliptic type: with constant Gauss curvature, with constant mean curvature, with parallel mean curvature vector field, with parallel normalized mean curvature vector field. The results obtained for the last class are used to give explicit solutions to the background systems of natural PDEs describing the timelike surfaces with parallel normalized mean curvature vector field in \(\mathbb {R}^4_1\) .
