TITLE: Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$
QUESTION [9 upvotes]: I am looking for positive integer solutions to the Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$ for distinct values of $(a,b,c,d)$.
There are many solutions with $a=b$ such as $(7,7,11,13)$ and $(13,13,22,23)$, and many solutions with $c=d$ such as $(3,5,7,7)$ and $(7,8,13,13)$, but it seems not so many with $4$ distinct values.  The smallest distinct solution I know of is $(35,139,146,169)$.
Are there infinitely many solutions?  Is there perhaps a parameterization of such solutions?
Of special interest are solutions where the quantity $(c^2+d^2)-(a^2+b^2)$ is a perfect square, such as $(a,b,c,d)=(323,392,407,713)$.  Are there other solutions of this type?
This comes up in solving an unsolved problem in geometry that I can't discuss, but you might get your name on a paper if you can help.

REPLY [3 votes]: $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)\tag{1}$
We show above equation gives infinitely many integer solutions if c = b+d.
We get
$$a^2 = \frac{c^2+d^2-b^2}{2}+\frac{\sqrt{-3c^4+6c^2d^2+18b^2c^2-3d^4+18b^2d^2-27b^4}}{6}\tag{2}$$
Hence we find the rational solutions of quartic curve for $c$ below $$v^2 = -3c^4+(18b^2+6d^2)c^2+18b^2d^2-27b^4-3d^4\tag{3}$$
Let $c = b+d$, then RHS of equation $(3)$ becomes to $-12b^2(b^2-2bd-2d^2)$.
Parametric solution of $b^2-2bd-2d^2=-3u^2$ is given
$(b,d)= \left( \dfrac{-5-2k+k^2}{(k+2)(k-1)}, \dfrac{2(1+k+k^2)}{(k+2)(k-1)} \right)$.
Hence equation $(3)$ has a solution $(c,v)= \left( \dfrac{7+4k+k^2}{(k+2)(k-1)}, \dfrac{6(k^2+4k+1)(-5-2k+k^2)}{(k+2)^2(k-1)^2} \right)$.
k is arbitrary.
Then we get $a^2 = \dfrac{3(k^4+4k^3+4k^2+3)}{(k+2)^2(k-1)^2}$.
Hence we find the rational solutions of quartic curve
$$V^2 = 3(k^4+4k^3+4k^2+3)\tag{4}$$
Quartic curve $(4)$ can be transformed to elliptic curve below.
$Y^2 = X^3 -156X + 560$ with rank 1 and generator is $P(X,Y)=(2,-16)$.
Hence this cuve has infinitely many rational points.
One of rational points is $2P(X,Y)=(\frac{65}{4}, \frac{-385}{8})$.
This point gives a solution $(a,b,c,d)=(1407459, 656995, 1391911, 2048906)$.
Thus equation $(1)$ gives infinitely many integer solutions.
Another solution of equation $(1)$ with $c\neq a\pm b, d\neq a\pm b$.
$a = 7n^4+100n^3+42n^2+100n+7$
$b = 13n^4+28n^3+174n^2+28n+13$
$c = 19n^4+112n^3+174n^2-56n+7$
$d = 7n^4-56n^3+174n^2+112n+19$
$n$ is arbitrary.
$n = 5$ is excluded.
             n     a      b      c       d
            [2    143    133    199     67]
            [3    247    217    373     61]
            [4   9271   8029  14599   1459]
            [6  32791  29341  54751   3931]
            [7   3367   3097   5761    433]
            [8   9263   8773  16207   1339]
            [9    481    469    859     79]
            [10 175207 175693 318847  32539]

Numerical solutions of equation $(1)$.
$(a,b,c,d)<1000$
fg1: 1 ($c=a\pm b,d=a\pm b$)
fg2: s ($c^2+d^2-a^2-b^2$ is square number)
There are many solutions such as $c\neq a\pm b, d\neq a\pm b$.
Total number of solutions : $378$
Case of $c=a\pm b,d=a\pm b: 274(72.5$ percecnt)
Others: $104(27.5$ percent)
There are two solutions such as $c^2+d^2-a^2-b^2$ is square number.
$(a,b,c,d)=(645, 323, 713, 407),(645, 392, 713, 407)$
There is no other solution with $(a,b,c,d)<10000$
                a     b     c     d fg1 fg2
                5     3     7     2 1   
                8     7    13     1 1   
               16     5    19    11 1   
               24    11    31    13 1   
               31    24    43    11     
               33     7    37    26 1   
               35    13    43    22 1   
               39    16    49    23 1   
               43    35    62    13     
               45    32    67    13 1   
               49    39    74    11     
               51    40    79    11 1   
               56    39    67    31     
               56     9    61    47 1   
               57    35    77    23     
               57    55    97     2 1   
               63    17    73    46 1   
               77    40   103    37 1   
               80    19    91    61 1   
               85    11    91    74 1   
               88    65   133    23 1   
               91    69   139    22 1   
               95    24   109    71 1   
               97    73   122    49     
              105   104   181     1 1   
              112    75   163    37 1   
              115    56   151    59 1   
              120    13   127   107 1   
              120    23   133    97 1   
              143   133   199    67     
              143    25   157   118 1   
              145   119   229    26 1   
              152    65   163   103     
              152    35   143   133     
              152   147   247    23     
              152    35   143   133     
              155    87   193    77     
              155   144   259    11 1   
              160    87   217    73 1   
              161    15   169   146 1   
              161   151   266    13     
              165    56   199   109 1   
              168    85   223    83 1   
              169    31   161   151     
              175    32   193   143 1   
              187    93   247    94 1   
              192   133   283    59 1   
              195    29   211   166 1   
              203    72   247   131 1   
              205   163   247   122     
              208    17   217   191 1   
              209   136   301    73 1   
              217    95   277   122 1   
              221    64   259   157 1   
              224    31   241   193 1   
              231   185   361    46 1   
              240   229   341   109     
              247   105   313   142 1   
              247   168   271   163     
              247   217   373    61     
              253   240   427    13 1   
              259   176   379    83 1   
              261    19   271   242 1   
              272   105   337   167 1   
              273   152   373   121 1   
              279    40   301   239 1   
              280   111   349   169 1   
              280   133   337   157     
              280   201   259   241     
              287   240   457    47 1   
              288    35   307   253 1   
              291   115   314   199     
              299   205   439    94 1   
              301   275   499    26 1   
              304   161   409   143 1   
              315    88   367   227 1   
              320    21   331   299 1   
              323    37   343   286 1   
              325   193   407   157     
              325   264   511    61 1   
              333   115   403   218 1   
              336   215   481   121 1   
              343   255   473   122     
              344   305   539    59     
              351   329   589    22 1   
              352   123   427   229 1   
              357    80   403   277 1   
              368   297   577    71 1   
              368   335   473   217     
              369   175   481   194 1   
              385    23   397   362 1   
              387   208   523   179 1   
              391   129   469   262 1   
              399    41   421   358 1   
              400   377   673    23 1   
              407    48   433   359 1   
              416   235   571   181 1   
              423   280   613   143 1   
              424   395   559   241     
              425   184   541   241 1   
              433   217   559   218     
              437    88   487   349 1   
              440    43   463   397 1   
              440   189   559   251 1   
              441   319   661   122 1   
              451   104   511   347 1   
              456    25   469   431 1   
              459   245   619   214 1   
              473   135   553   338 1   
              475   141   559   334 1   
              477   403   763    74 1   
              480   371   739   109 1   
              493   312   703   181 1   
              495   448   817    47 1   
              504   409   581   337     
              513   200   637   313 1   
              517   203   643   314 1   
              520   341   751   179 1   
              520   147   607   373 1   
              527   385   793   142 1   
              528    47   553   481 1   
              533    27   547   506 1   
              536   465   781   169     
              539   240   691   299 1   
              544   531   931    13 1   
              545   441   806   139     
              551   265   721   286 1   
              552   145   637   407 1   
              559    56   589   503 1   
              559   240   671   329     
              560   429   859   131 1   
              561   464   889    97 1   
              575    49   601   526 1   
              583   320   793   263 1   
              589   336   811   253 1   
              589   555   946    91     
              595   549   991    46 1   
              600   217   733   383 1   
              611   120   679   491 1   
              613   245   598   473     
              616    29   631   587 1   
              616   159   709   457 1   
              616   559   889   229     
              621   104   679   517 1   
              624   325   851   301     
              627   413   907   214 1   
              637   155   727   482 1   
              645   323   713   407    s
              645   392   713   407    s
              649   455   961   194 1   
              665   336   671   481     
              665   193   767   473     
              667   165   763   502 1   
              675    53   703   622 1   
              679   151   722   539     
              680   427   967   253 1   
              689   231   829   458 1   
              695   488   817   403     
              696   385   949   311 1   
              704   295   889   409 1   
              705    31   721   674 1   
              713   232   853   481 1   
              715   272   883   443 1   
              721   344   923   379     
              725   112   787   613 1   
              728    55   757   673 1   
              728   165   823   563 1   
              728   703   937   469     
              735    64   769   671 1   
              744   169   869   581     
              759   305   949   454 1   
              767   360   997   407 1   
              775   177   877   598 1   
              776   489   869   481     
              795   136   871   659 1   
              800    33   817   767 1   
              805   189   869   626     
              825   248   973   577 1   
              832   183   937   649 1   
              840    59   871   781 1   
              845   328   763   713     
              883   480   893   637     
              899    61   931   838 1   
              901    35   919   866 1   
              907   155   938   767     
              912   427   973   613     
              921    91   913   851     
              935    72   973   863 1   
              936   775   989   739