I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me how I can do it? I already solved this system with Mathematica but I would like to redo these calculations by hand to understand how their were obtained.
The system is: $\left\{ \begin{array}{l l} 4u - 3v + 4w + 3z = 1\\ -4v - 3u - 4z + 3w = 0 \end{array} \right.$
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There is no way, as I know, to determine all integer solutions to the system. You should at first find all real solutions to the system, this gives you a subspace $V\subset\mathbb R^m$ of dimension at least $m-n$. Now for finding all integer solutions you should find the lattice points of the subspace $V$.
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Well, finally I have used old good Hermite Normal Form. I knew this method before but I thought about something which could be easier for bigger systems to solve by hand.
If something check paper Linear Diophantine Equations, W. J. Gilbert, A. Pathria, 1990.
What they used is just unimodular row operations to obtain HNF and matrix U at once. In my case I had to work on symbol coefficients and this approach led me to all what I needed. Well, if you know any better way I will be happy to give it a try. I know about Smith Normal Form but it sounds to complex - well, I have to say that I have never tried to us SNF. Maybe I should.
A linear equation with integer coefficients, where one looks for integer solutions is called a linear Diophantine equation.
The simplest case $$ a x + b y = c $$ can be solved systematically and has either no or infinite many solutions.
From here one can move to more variables or more equations.
See System of linear Diophantine equations on how you might proceed. It recommends calculating the Smith normal form.