ax^4 + by^4 + cz^4 + dw^4 = 0, abcd=□

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
Furthermore, condition of a + b + c + d = 0 or having one known solution is necessary to have a rational solution.
We show some parametric solutions of ax^4 + by^4 + cz^4 + dw^4 = 0 with abcd is square number.

General case:

     4(m^2-1)X^4 + (1-m^2)(m^2+3)Y^4 = -(m^2-1)^2Z^4 + 4(m^2+3)W^4
     
     (d^2-2dm^2+2m^4)X^4 + (d^2-2dm^2+2m^4)Y^4 = 2(-d+m^2)^2Z^4 + 2W^4
     
     -dm^2(d+2)X^4 + (-2m^2+d+2)m^2(d-2m^2)Y^4 = d(d-2m^2)m^4Z^4 + (d+2)(2m^2-d-2)W^4
     
     (m^4+1)X^4 + (m^4+1)Y^4 = 2Z^4 + 2W^4
     
     2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^4
     
     pX^4 + pY^4 = qZ^4 + qW^4
     
Individual case:

According Richmond' theorem, since below solutions have rational solutions then we can obtain infinitely many integer solutions.
We show only small solutions with {x,y,z,w}<1000.
Equation ax^4 + by^4 = cz^4 + dw^4 with a+b-c = 16d has smallest non zero solution ( x, y, z, w)=( 1, 1, 1, 2).
a<10,{b,c,d}<50, a+b≠c+d.

[ a  b  c  d] [ x  y  z  w]

[1, 20, 5, 1][1, 1, 1, 2]
[1, 20, 5, 1][19, 5, 13, 2]
[1, 20, 5, 1][37, 19, 5, 46]
[1, 20, 5, 1][199, 5, 133, 44]
[1, 20, 5, 1][261, 112, 34, 297]
[1, 20, 5, 1][821, 509, 661, 958]
[1, 20, 5, 1][881, 139, 573, 516]
[1, 27, 12, 1][1, 1, 1, 2]
[1, 27, 12, 1][23, 9, 1, 26]
[1, 27, 12, 1][167, 57, 97, 26]
[1, 49, 18, 2][1, 1, 1, 2]
[1, 49, 18, 2][73, 23, 39, 22]
[1, 49, 18, 2][919, 253, 458, 497]
[2, 18, 4, 1][1, 1, 1, 2]
[2, 18, 4, 1][13, 3, 11, 2]
[2, 18, 4, 1][23, 1, 19, 14]
[2, 18, 4, 1][47, 33, 23, 74]
[2, 18, 4, 1][79, 57, 83, 94]
[2, 18, 4, 1][107, 283, 179, 578]
[2, 18, 4, 1][139, 29, 1, 166]
[2, 18, 4, 1][247, 289, 419, 314]
[2, 18, 4, 1][431, 369, 527, 554]
[2, 28, 14, 1][1, 1, 1, 2]
[2, 28, 14, 1][17, 7, 1, 22]
[2, 28, 14, 1][29, 11, 19, 2]
[2, 28, 14, 1][137, 13, 33, 162]
[2, 28, 14, 1][173, 243, 269, 402]
[2, 28, 14, 1][207, 77, 135, 50]
[3, 25, 12, 1][1, 1, 1, 2]
[3, 25, 12, 1][3, 1, 1, 4]
[3, 25, 12, 1][9, 131, 157, 92]
[3, 25, 12, 1][23, 9, 17, 6]
[3, 25, 12, 1][151, 73, 31, 218]
[3, 25, 12, 1][551, 185, 399, 202]
[3, 49, 4, 3][1, 1, 1, 2]
[3, 49, 4, 3][1, 31, 41, 58]
[3, 49, 4, 3][1, 417, 153, 838]
[3, 49, 4, 3][3, 2, 4, 1]
[3, 49, 4, 3][7, 4, 2, 9]
[3, 49, 4, 3][21, 1, 19, 12]
[3, 49, 4, 3][27, 47, 77, 76]
[3, 49, 4, 3][29, 3, 27, 2]
[3, 49, 4, 3][193, 127, 17, 274]
[3, 49, 4, 3][223, 193, 167, 394]
[3, 49, 4, 3][239, 47, 223, 82]
[3, 49, 4, 3][329, 48, 306, 103]
[3, 49, 4, 3][701, 259, 659, 502]
[5, 20, 9, 1][1, 1, 1, 2]
[5, 20, 9, 1][17, 7, 15, 10]
[5, 20, 9, 1][61, 299, 163, 626]
[5, 20, 9, 1][71, 49, 39, 122]
[5, 20, 9, 1][335, 265, 273, 578]
[5, 20, 9, 1][385, 337, 383, 646]
[5, 20, 9, 1][529, 215, 433, 586]
[5, 30, 3, 2][1, 1, 1, 2]
[5, 30, 3, 2][7, 4, 9, 1]
[5, 30, 3, 2][15, 1, 17, 6]
[5, 30, 3, 2][47, 65, 79, 122]
[5, 30, 3, 2][101, 171, 69, 338]
[5, 30, 3, 2][153, 10, 89, 189]
[5, 30, 3, 2][177, 83, 209, 132]
[5, 30, 3, 2][447, 34, 151, 561]
[5, 45, 18, 2][1, 1, 1, 2]
[5, 45, 18, 2][1, 5, 6, 7]
[5, 45, 18, 2][3, 1, 2, 3]
[5, 45, 18, 2][3, 157, 179, 258]
[5, 45, 18, 2][9, 11, 14, 3]
[5, 45, 18, 2][11, 3, 1, 14]
[5, 45, 18, 2][11, 7, 10, 5]
[5, 45, 18, 2][15, 1, 7, 18]
[5, 45, 18, 2][21, 11, 5, 30]
[5, 45, 18, 2][37, 15, 18, 47]
[5, 45, 18, 2][45, 37, 47, 54]
[5, 45, 18, 2][119, 41, 89, 22]
[5, 45, 18, 2][123, 119, 22, 267]
[5, 45, 18, 2][157, 1, 86, 179]
[5, 45, 18, 2][169, 93, 14, 247]
[5, 45, 18, 2][175, 87, 69, 242]
[5, 45, 18, 2][261, 175, 242, 207]
[5, 45, 18, 2][279, 169, 247, 42]
[5, 45, 18, 2][339, 443, 545, 570]
[5, 45, 18, 2][367, 355, 426, 571]
[5, 45, 18, 2][421, 161, 190, 535]
[5, 45, 18, 2][443, 113, 190, 545]
[5, 45, 18, 2][483, 421, 535, 570]
[5, 45, 18, 2][701, 313, 238, 943]
[5, 45, 18, 2][721, 275, 18, 947]
[5, 45, 18, 2][787, 705, 922, 341]
[5, 45, 18, 2][825, 721, 947, 54]
[5, 45, 18, 2][883, 245, 649, 254]
[5, 45, 18, 2][899, 515, 771, 434]
[5, 45, 18, 2][907, 237, 665, 230]
[5, 45, 18, 2][939, 701, 943, 714]
[6, 12, 2, 1][1, 1, 1, 2]
[6, 12, 2, 1][3, 7, 11, 2]
[6, 12, 2, 1][7, 1, 9, 6]
[6, 12, 2, 1][7, 17, 23, 26]
[6, 12, 2, 1][29, 43, 13, 82]
[6, 12, 2, 1][251, 359, 563, 386]
[6, 12, 2, 1][329, 277, 513, 222]
[6, 12, 2, 1][331, 113, 269, 502]
[6, 27, 1, 2][1, 1, 1, 2]
[6, 27, 1, 2][3, 1, 1, 4]
[6, 27, 1, 2][19, 3, 29, 14]
[6, 27, 1, 2][23, 153, 263, 266]
[6, 27, 1, 2][29, 207, 337, 368]
[6, 27, 1, 2][73, 11, 101, 76]
[6, 27, 1, 2][189, 127, 257, 268]
[6, 30, 20, 1][1, 1, 1, 2]
[6, 30, 20, 1][11, 5, 3, 18]
[6, 30, 20, 1][11, 211, 193, 422]
[6, 30, 20, 1][17, 127, 137, 166]
[6, 30, 20, 1][31, 17, 25, 22]
[6, 39, 13, 2][1, 1, 1, 2]
[6, 39, 13, 2][3, 1, 1, 4]
[6, 39, 13, 2][9, 1, 7, 8]
[6, 39, 13, 2][9, 221, 203, 434]
[6, 39, 13, 2][11, 1, 9, 6]
[6, 39, 13, 2][29, 19, 13, 46]
[6, 39, 13, 2][29, 149, 187, 202]
[6, 39, 13, 2][41, 11, 27, 48]
[6, 39, 13, 2][51, 19, 43, 28]
[6, 39, 13, 2][61, 19, 51, 18]
[6, 39, 13, 2][71, 3, 29, 92]
[6, 39, 13, 2][73, 147, 179, 224]
[6, 39, 13, 2][153, 499, 653, 412]
[6, 39, 13, 2][173, 11, 59, 226]
[6, 39, 13, 2][177, 21, 53, 232]
[6, 39, 13, 2][449, 23, 249, 558]
[6, 39, 13, 2][449, 341, 43, 788]
[7, 21, 12, 1][1, 1, 1, 2]
[7, 21, 12, 1][19, 13, 11, 34]
[7, 21, 12, 1][21, 11, 19, 18]
[7, 35, 10, 2][1, 1, 1, 2]
[7, 35, 10, 2][25, 7, 23, 10]
[7, 35, 10, 2][53, 43, 37, 94]
[7, 35, 10, 2][91, 101, 123, 174]
[7, 35, 10, 2][177, 183, 259, 154]
[7, 35, 10, 2][211, 11, 193, 22]
[7, 35, 10, 2][277, 205, 77, 476]
[7, 35, 10, 2][353, 23, 309, 306]
[8, 25, 1, 2][1, 1, 1, 2]
[8, 25, 1, 2][1, 23, 41, 38]
[8, 25, 1, 2][19, 85, 149, 142]
[8, 25, 1, 2][47, 41, 23, 86]
[8, 25, 1, 2][107, 19, 173, 94]
[8, 25, 1, 2][211, 355, 797, 262]
[8, 25, 1, 2][389, 491, 661, 922]
[8, 27, 3, 2][1, 1, 1, 2]
[8, 27, 3, 2][13, 27, 37, 46]
[8, 27, 3, 2][53, 3, 67, 34]
[8, 27, 3, 2][131, 161, 255, 250]
[8, 27, 3, 2][221, 519, 743, 854]
[8, 27, 3, 2][463, 447, 833, 154]
[8, 49, 25, 2][1, 1, 1, 2]
[8, 49, 25, 2][29, 41, 49, 4]
[8, 49, 25, 2][79, 41, 23, 122]
[8, 49, 25, 2][163, 277, 299, 466]
[8, 49, 25, 2][193, 69, 147, 128]
[8, 49, 25, 2][449, 311, 265, 758]
[8, 49, 25, 2][533, 227, 419, 226]
[9, 25, 2, 2][1, 1, 1, 2]
[9, 25, 2, 2][33, 1, 26, 47]
[9, 25, 2, 2][111, 433, 671, 698]
[9, 25, 2, 2][603, 229, 634, 827]
[9, 46, 23, 2][1, 1, 1, 2]
[9, 46, 23, 2][41, 11, 23, 56]
[9, 46, 23, 2][51, 29, 19, 82]
[9, 46, 23, 2][123, 53, 101, 62]