What is the best algorithm to find all the integer points (X,Y) on this curve
$X^3+aXbY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)?
What is the best algorithm to find all the integer points (X,Y) on this curve $X^3+aXbY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)? 


Strictly speaking, the only algorithm of which I'm aware for solving such cubic Thue equations goes via lower bounds for linear forms in complex logarithms (and lattice basis reduction) and is described in, for example, a paper of Tzanakis and de Weger [JNT 31 (1989), 99132]. Approaches based on Skolem's $p$adic method or on lower bounds for linear forms in elliptic logarithms should typically work here too. 

