Ajai Choudhry has given a general solutiona general solution for a cube represented as sum of three positive cubes.
Where:
$d^3=a^3+b^3+c^3$ ----(1)
The link to his paper is given below.
file:///C:/Users/User/Downloads/1181071714.pdf
His solution given in equation (14) of the paper is valid for all positive integers (a,b,c)$(a,b,c)$ where a>b$a>b$. Since there are infinite positive integers (a,binfinitely many such triples,c) of which there are infinite integers which are (a>b) It it is easy to see then that your equation (1) above will have infiniteinfinitely many solutions.
