  For a complete minimal surface in the Euclidean 3-space, the so-called flux
vector corresponds to each end. The flux vectors are balanced, i.e., the sum of
those over all ends are zero. Consider the following inverse problem: For each
balanced n vectors, find an n-end catenoid which attains given vectors as flux.
Here, an n-end catenoid is a complete minimal surface of genus 0 with ends
asymptotic to the catenoids. In this paper, the problem is reduced to solving
algebraic equation. Using this reduction, it is shown that, when n=4, the
inverse problem for 4-end catenoid has solutions for almost all balanced 4
vectors. Further obstructions for n-end catenoids with parallel flux vectors
are also discussed.
