We show the parametric solutions  of x^6 + az^6 = y^6 + aw^6.
Since there are many solutions, only a part of the solutions are shown.


   Table of solutions  of x^6 + az^6 = y^6 + aw^6.
         a                             x      z      y       w ] 
[p^6+q^6,                            p^2,    q^2,    q,      p ]

[8(n^2+27)(n^4+198n^2+9),         n^2+8n+3, n-1, n^2-8n+3,  n+1]

[18(n^2+1)(27n^4+58n^2+27),      3n^2+2n+3, n-1, 3n^2-2n+3, n+1]

[4(3n^2+5)(27n^2+25)(3n^2+25),   3n^2+4n+5, n-1, 3n^2-4n+5, n+1]

[8(27n^2+1)(9n^4+198n^2+1),      3n^2+8n+1, n-1, 3n^2-8n+1, n+1]

[72(n^2+27)(27n^4+226n^2+243),   3n^2+8n+9, n-1, 3n^2-8n+9, n+1]

[4(25n^2+27)(5n^2+3)(25n^2+3),   5n^2+4n+3, n-1, 5n^2-4n+3, n+1]

[1152(n^2+1)(27n^4+58n^2+27),    6n^2+4n+6, n-1, 6n^2-4n+6, n+1]

[13122(n^2+1)(27n^4+58n^2+27),   9n^2+6n+9, n-1, 9n^2-6n+9, n+1]

[72(27n^2+1)(243n^4+226n^2+27),  9n^2+8n+3, n-1, 9n^2-8n+3, n+1]

[8(9n^4-21n^2+64)(27n^6+18n^4+3n^2+64),    3n^3+n+8,  n-1, 3n^3+n-8,  n+1]

[4(3n^2+5)(9n^4+3n^2+16)(9n^4+27n^2+16),   3n^3+5n+4, n-1, 3n^3+5n-4, n+1]

[2(3n^2+7)(3n^2+4)(27n^6+126n^4+147n^2+4), 3n^3+7n+2, n-1, 3n^3+7n-2, n+1]

[18(3n^2+7)(3n^2+4)(9n^6+42n^4+49n^2+108), 3n^3+7n+6, n-1, 3n^3+7n-6, n+1]

[72(9n^4+51n^2+64)(3n^6+18n^4+27n^2+64),   3n^3+9n+8, n-1, 3n^3+9n-8, n+1]

n,p,q  are arbitrary.