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.
So, we are curious about whether above equation has a solution or not if abcd is not square number.
In particular, when does this equation have infinitely many integer solutions?

General case:

     pX^4 + pY^4 = 2Z^4 + W^4
     
     pX^4 + pY^4 + 2Z^4 = sW^4
     
     d^2a^2X^4 + (-d+2a)^2a^2Y^4 + 2d(-d+2a)Z^4 = 4a^2W^4
     
     pX^4 + pY^4 = 2Z^4 + W^4 revisited
     
     pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)W^4 revisited
     
     (d-2)(d+2)^2X^4 + (4d-8)Y^4 + Z^4 = (4d+8)W^4
     
     (m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4
     
     px^4 + py^4 = 2pu^4 + qv^4
     
     (-c^2+4ec-2e^2)X^4 + (-c^2+4ec-2e^2)Y^4 + (2c-2e)^2Z^4 = 2W^4

      d^2X^4 + (d-2c)^2Y^4 = 2d(d-2c)Z^4 + 4W^4

     
Individual case:

     5X^4 + 45Y^4 + Z^4 = 6W^4  

     2X^4 + 8Y^4 + Z^4 = 9W^4  

     31X^4 + 31Y^4 + Z^4 = 2W^4  

     7X^4 + 7Y^4 + 2Z^4 = 9W^4  

     25X^4 + 4Y^4 + Z^4 = 5W^4  

     79X^4 + 79Y^4 + 4Z^4 = 2W^4  

     57X^4 + 57Y^4 = 8Z^4 + W^4 

     18X^4 + 8Y^4 + Z^4 = 27W^4 

     73X^4 + 73Y^4 + Z^4 = 2W^4 

     75X^4 + 3Y^4 + 2Z^4 = 5W^4 

     33X^4 + 33Y^4 = 2Z^4 + W^4 

     3X^4 + 3Y^4 = 2Z^4 + W^4 

     9X^4 + 4Y^4 + 2Z^4 = 6W^4 

     25X^4 + 25Y^4 + 8Z^4 = 25W^4 

     X^4 + 9Y^4 + 6Z^4 = W^4

     9X^4 + 36Y^4 + 4Z^4 = 9W^4 

     7X^4 + 7Y^4 + Z^4 = 8W^4 

     47X^4 + 47Y^4 + 2Z^4 = 49W^4 

     23X^4 + 23Y^4 + 2Z^4 = 25W^4 

     X^4 + 36Y^4 + 27Z^4 = 4W^4 

     49X^4 + 49Y^4 + 8Z^4 = W^4 
     
     3X^4 + 75Y^4 + 5Z^4 = 8W^4 

     36X^4 + Y^4 + 3Z^4 = W^4 

     25X^4 + Y^4 + 10Z^4 = W^4 

     50X^4 + 2Y^4 + 5Z^4 = 2W^4 

     7X^4 + 12Y^4 = 4Z^4 + 3W^4 

     4X^4 + Y^4 + Z^4 = W^4 

     31X^4 + 31Y^4 + 2Z^4 = W^4 

     12X^4 + 12Y^4 = 8Z^4 + W^4 

     19X^4 + 19Y^4 = 2Z^4 + W^4 

     17X^4 + 17Y^4 = 18Z^4 + W^4 

     9X^4 + Y^4 + 2Z^4 = 3W^4 

     12X^4 + 3Y^4 + Z^4 = W^4 

     9X^4 + 4Y^4 + 3Z^4 = W^4 

     9X^4 + 9Y^4 = 2Z^4 + W^4