Higher-dimensional PT-symmetric potentials constituted by delta-sign-exponential (DSE) functions are created in order to show that the exceptional points in the non-Hermitian Hamiltonian can be converted to those in the corresponding one-dimensional (1D) geometry, no matter the potentials inside are rotationally symmetric or not. These results are first numerically observed and then are proved by mathematical methods. For spatially varying Kerr nonlinearity, 2D exact peakons are explicitly obtained, giving birth to families of stable square peakons in the rotationally symmetric potentials and rhombic peakons in the nonrotationally symmetric potentials. By adiabatic excitation, different types of 2D peakons can be transformed stably and reciprocally. Under periodic and mixed perturbations, the 2D stable peakons can also travel stably along the spatially moving potential well, which implies that it is feasible to manage the propagation of the light by regulating judiciously the potential well. However, the vast majority of high-order vortex peakons are vulnerable to instability, which is demonstrated by the linear-stability analysis and by direct numerical simulations of propagation of peakon waveforms. In addition, 3D exact and numerical peakon solutions including the rotationally symmetric and the nonrotationally symmetric ones are obtained, and we find that incompletely rotationally symmetric peakons can occur stably in completely rotationally symmetric DSE potentials. The 3D fundamental peakons can propagate stably in a certain range of potential parameters, but their stability may get worse with the loss of rotational symmetry. Exceptional points and exact peakons in n dimensions are also summarized.