Are there more than two rational solutions to a certain system $abcd=a+b+c+d=K$ ($K$ a given constant)?

Question

This question is a follow-on of a previous question asked some days ago which has been deleted due to its lack of precision. In fact, I found it well explained here

But, implicitly, the domain of solutions was numbers with at most two decimals.

Let me restate the question with rational numbers :

Find four rational numbers (it is the domain I am interested in) $a \le b \le c \le d$ whose product is equal to their sum, this common value being $7.11$ :

$$abcd=a+b+c+d=\frac{711}{100}$$

The solution provided, converted into fractional expressions, is :

$$\begin{cases}a&=&6/5&=&1.20\\b&=&5/4&=&1.25\\c&=&3/2&=&1.50\\d&=&79/25&=&3.16\end{cases}$$

The numbers involved aren't so surprizing because of the prime factor decompositions :

$$711=3^2 \times 79 \ \ \text{and} \ \ 100=2^2 \times 5^2.$$

Using a computer "screening" and the above decompositions, I have found another solution :

$$\begin{cases}a&=&81/100&=&0.81\\b&=&5/3&=&1.666...\\c&=&2&&\\d&=&79/30&=&2.6333...\end{cases}$$

My questions : are there other (rational) solutions and how can we find them ? Beyond that, is there some theory behind (like the theory of rational points on cubic curves) ?

Answer 1

$$abcd=a+b+c+d=K$$

Substitute $d=K-a-b-c$ into $abcd=K,$ we get $$-abc^2+ab(K-a-b)c-K=0$$ For the quadratic in $c$ to have rational solutions, the discriminant must be a rational square. Hence, this problem is reduced to the problem of finding the rational solutions for the quartic equation. $$v^2 = a^4b^2+b(2b^2-2bK)a^3+b(bK^2+b^3-2b^2K)a^2-4abK$$ b and K are given rational numbers.

               (a,b,c,d), K=711/100

      158/205, 75/41, 41/20, 123/50 
      237/200, 6/5, 8/5, 25/8
      252/325, 7/4, 395/182, 169/70
      243/325, 79/39, 25/12, 169/75 
      15/19, 237/133, 49/25, 361/140 
      16/15, 121/100, 81/44, 395/132 
      395/448, 48/35, 147/64, 64/25 
      21/23, 161/100, 23/14, 474/161 
      75/98, 474/245, 49/25, 49/20 
      1185/1219, 6/5, 2809/1150, 529/212
      22/25, 11/8, 632/275, 225/88 
      15/14, 79/63, 243/140, 686/225 
      77/100, 21/11, 150/77, 869/350 
      1343/1600, 25/17, 153/64, 1024/425 
      207/275, 1975/1012, 23/11, 1331/575 
      1369/1400, 6/5, 600/259, 3871/1480
      8/9, 158/117, 169/72, 6561/2600 
      2844/2465, 841/680, 1156/725, 25/8 
      539/600, 11/7, 400/231, 6399/2200 
      28/33, 2765/1914, 841/350, 9801/4060 
      20/19, 20449/16150, 85557/48620, 867/286 
      85557/100100, 10/7, 24843/10450, 605/247