On Homogeneous Cubic Equation with Four Unknowns

The cubic diophantine equation with four unknowns given by     3 3 2 2 x  y  x  y x  y 16zw is
analyzed for its non-zero distinct integer points on it. Different patterns of integer points for the equation
under consideration are obtained. A few interesting relations between the solutions and special number
patterns namely Polygonal number, Gnomonic number, Star number and Pronic number are presented.