We investigate the $E\ensuremath{\bigotimes}e$ Jahn-Teller (JT) system when both linear and quadratic couplings are included and when it is subjected to external symmetry-lowering distortions. Such distortions could, for example, be due to intrinsic molecular geometry, strains applied to a cubic crystal, and as the result of cooperative interactions between different JT centers. It is well known that the lowest adiabatic potential-energy surface consists of a warped trough containing up to three minima depending on competition between the symmetry-lowering distortion and the quadratic coupling. The motion of the system can be divided into a vibration across the trough and a hindered pseudorotation around the trough. We will construct an analytical form for the wave function involving electronic, vibrational, and pseudorotational contributions that is valid for a wide range of coupling strengths. The wave function is then used to calculate the energy of the ground state of the system. The results are compared to those from a numerical Lanczos diagonalization procedure. We find that in strong linear coupling there tends to be very little dependence on either the strength of the distortion or the value of the quadratic coupling constant. This is not true for weaker linear coupling. We show that much of the behavior of the system can be deduced from analyzing the form of the potential, so that it is not necessary to perform a complete analysis of the wave functions and their energies.