Consider the following system.
1) $a_{11}x_1 + a_{21}x_2 + \cdots + a_{m1}x_m=d_1$
2) $a_{12}x_1 + a_{22}x_2 + \cdots + a_{m2}x_m=d_2$
$\vdots$
n) $a_{1n}x_1 + a_{2n}x_2 + \cdots + a_{mn}x_m=d_n$
$\bullet$ $x_i\leq M_i$, for each $i=1, \cdots, m$.
For every $i=1, 2, \cdots, m $ and $j= 1, 2, \cdots, n$, we know that $a_{ij}$ is equal to $-1,\ 0$, or $1$. All the right hand side amounts, $d_j$s, are zero exception of two of them which are nonzero and equal. Moreover, for each $i=1,\cdots, m$, $M_i$ is a given integer number.
I need an algorithm to solve and find all the integer solutions of the system. As I need to implement the algorithm, it would be nicer the algorithm being more practical and easy to implement.
This is a linear system; as explained here, systems of linear diophantine equations can be solved by finding their Smith normal form or Hermite normal form. (The first link explains how to use the Smith normal form in detail. Using the Hermite normal form is analogous. An online calculator for the Hermite method, using the "LLL-based algorithm of Havas-Majewski-Matthews" is here.) This question at StackOverflow indicates that Octave and Octave embedded can do what you want. Another SO question has an answer pointing to a python implementation of the Hermite/LLL/Havas-Majewski-Matthews solution, Diophantine.