Decomposition of an integer as the sum of two cubes to a fixed modulus

The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime integer N there is given an inductive way to find its remainders that can be represented as the sum of two cubes to a fixed modulus N. Moreover, it is possible to find the components of this representation.