The current study aims to present a novel design of a sixth-order (SO) nonlinear Emden–Fowler nonlinear system (SO-NSEFM) along with its five types. The novel design of SO-NSEFM is achieved using the typical second-order Emden–Fowler system. The detail of the singularity and shape factors is presented for each type of the SO-NSEFM. Three different examples of each type of the designed SO-NSEFM will be solved using the supervised neural network (SNN) Levenberg–Marquardt backpropagation approach (LMBA), i.e., SNN–LMBA. A reference dataset using the spectral collocation scheme with the proposed SNN–LMBA will be established for the designed SO-NSEFM. The achieved approximate outcomes of the designed SO-NSEFM are accessible using the procedures of testing, verification, and training of the proposed neural networks to reduce the MSE. For the efficiency, correctness, and effectiveness of the proposed SNN-LMBA, the investigations are presented through the proportional performances of regression, MSE results, correlation and error histograms (EHs), and regression.