ax^3+by^3+cz^3 = dxyz 


If diophantine equation ax^3+by^3+cz^3 = dxyz has a integer solution, then we can obtain a new solution.
Hence we can obtain infinitely many integer solutions by repeating this process.

ax^3+by^3+cz^3 = dxyz............................................................................(1)

Assuming (x,y,z)=(u,v,w) is a known solution.

Let x=pt+u, y=t+v, z=t+w, then  equation (1) reduces to

(c+ap^3-dp+b)t^3+(-dpw+3ap^2u-du+3cw-dpv+3bv)t^2+(-duw-duv+3apu^2+3cw^2-dpvw+3bv^2)t = 0.........(2)

Let p = (duw+duv-3cw^2-3bv^2)/(3au^2-dvw) and 

t = (d^4w^4uv^2+d^4w^3uv^3-3d^3w^5cv^2-81a^2u^3c^2w^4-81a^2u^3b^2v^4-81cwa^3u^6+d^4v^4uw^2
    -3d^3v^5bw^2-81bva^3u^6+27du^7a^3-27d^2u^5a^2vw-3d^3w^3u^3va+3d^3w^2u^3v^2a+27auc^2w^5dv
    +27aub^2v^5dw+108cw^2a^2u^4dv-3d^3v^3u^3wa+108bv^2a^2u^4dw+27dw^3ca^2u^4-27d^2w^2bv^3au^2
    -27au^2d^2v^2cw^3+27a^2u^4dv^3b-162a^2u^3cw^2bv^2+54aucw^3bv^3d)
    /(-cd^3v^3w^3+27ca^3u^6-27ca^2u^4dvw+9ad^3u^3w^2v-9ad^2u^2w^4c+9ad^3u^3wv^2-27ac^3w^6
    -27ab^3v^6+27ba^3u^6+27aduw^5c^2-9ad^2u^2v^4b+27aduv^5b^2-81ac^2w^4bv^2-81acw^2b^2v^4
    +ad^3u^3w^3+ad^3u^3v^3-9d^2u^5wa^2-d^4uw^3v^2-9d^2u^5va^2-d^4uv^3w^2+3cw^4d^3v^2+3bv^4d^3w^2
    -bd^3v^3w^3-36ad^2u^2w^3vc-36ad^2u^2wv^3b+54aduw^3cbv^2+27aduwb^2v^4+27aduvc^2w^4
    +54aduv^3cw^2b+27dcw^2a^2u^4+27dbv^2a^2u^4-27ba^2u^4dvw).
        
Then we obtain a new solution of equation (1) below.

x = -9d^2vw^6c^2-9d^2v^6wb^2-3u^2d^4w^3v^2-3u^2d^4v^3w^2-u^2d^4v^4w-u^2d^4w^4v+81a^2u^4b^2v^3
    +81a^2u^4c^2w^3+au^4d^3w^3+au^4d^3v^3+54auc^3w^6+54aub^3v^6-81au^2dwb^2v^4-27au^2dv^5b^2
    -108au^2dv^3cw^2b-108au^2dw^3cbv^2+162auc^2w^4bv^2+162aucw^2b^2v^4-54a^2u^5cdvw-27a^2u^5dcw^2
    -27a^2u^5dbv^2-54a^2u^5bdvw+81a^2u^4cw^2bv+81a^2u^4bv^2cw+6ubv^5d^3w+6ucw^5d^3v+9ucw^4d^3v^2
    +9ubv^4d^3w^2+2ucd^3v^3w^3+2ubd^3v^3w^3-9d^2w^4cbv^3-9d^2v^4cw^3b+3au^4d^3w^2v+3au^4d^3wv^2
    +27au^3d^2v^2cw^2+18au^3d^2wv^3b+18au^3d^2w^3vc+27au^3d^2w^2bv^2-81au^2dvc^2w^4-27au^2dw^5c^2+27a^3u^7c+27a^3u^7b

y = -27ca^2u^4dv^2w-27ab^3v^7+d^4w^4uv^2-3d^3w^5cv^2-81a^2u^3c^2w^4-81a^2u^3b^2v^4-81cwa^3u^6
    -54bva^3u^6+27du^7a^3-36d^2u^5a^2vw-2d^3w^3u^3va+12d^3w^2u^3v^2a+54auc^2w^5dv+54aub^2v^5dw
    +135cw^2a^2u^4dv+6d^3v^3u^3wa+81bv^2a^2u^4dw+27dw^3ca^2u^4-27d^2w^2bv^3au^2-63au^2d^2v^2cw^3
    +54a^2u^4dv^3b-162a^2u^3cw^2bv^2+108aucw^3bv^3d-bd^3v^4w^3+27vca^3u^6+3cw^4d^3v^3-9d^2u^5v^2a^2
    -81acw^2b^2v^5+27aduv^6b^2-81ac^2w^4bv^3-9ad^2u^2v^5b-cd^3v^4w^3+ad^3u^3v^4-27vac^3w^6-9vad^2u^2w^4c
    +54aduv^4cw^2b+27aduv^2c^2w^4-36ad^2u^2wv^4b

z = 54aduw^4cbv^2+27aduw^2b^2v^4-27ba^2u^4dvw^2+27wba^3u^6-9ad^2u^2w^5c-bd^3v^3w^4-27wab^3v^6-9d^2u^5w^2a^2
   +27aduw^6c^2-81ac^2w^5bv^2-81acw^3b^2v^4-27ac^3w^7+ad^3u^3w^4-81a^2u^3c^2w^4-81a^2u^3b^2v^4-54cwa^3u^6
   +d^4v^4uw^2-3d^3v^5bw^2-81bva^3u^6+27du^7a^3-36d^2u^5a^2vw+6d^3w^3u^3va+12d^3w^2u^3v^2a+54auc^2w^5dv
   +54aub^2v^5dw+81cw^2a^2u^4dv-2d^3v^3u^3wa+135bv^2a^2u^4dw+54dw^3ca^2u^4-63d^2w^2bv^3au^2-27au^2d^2v^2cw^3
   +27a^2u^4dv^3b-162a^2u^3cw^2bv^2+108aucw^3bv^3d+3bd^3v^4w^3-cw^4d^3v^3-36vad^2u^2w^4c-9ad^2u^2wv^4b

a,b,c are arbitrary.


Example: 3x^3 + 2y^3 + 2z^3 = 8xyz

(a,b,c,d)=(3, 2, 2, 8), (u,v,w)=(14, 19, 25)

New solution is (x,y,z)=(-122724, 218671, -68575).