Observations on the biquadratic with five unknows x⁴ - y⁴ - 2xy(x² - y²) = z(X² + Y²)

We obtain infinitely many non-zero integer quintuples (x, y, z, X, Y) satisfying the biquadratic equation with five unknowns x⁴ - y⁴ - 2xy(x² - y²) = z(X² + Y²). Various interesting properties between the values of x, y, z, X, Y and special number patterns, namely, polygonal numbers, centered pyramidal and polygonal numbers, Jacob-lucas numbers and kynea numbers are presented.