Maybe I'm confused with something I learned on a Linear Programming course, but:
Is it true that a linear homogeneous system of equations with integer coefficients and at least one positive real solution also has an integer positive solution?
If true, do you know about a reference for this result?
A basis of the vector space that represents the solution of a linear homogeneous system of equations (with $n$ variables) can be found by using elementary arithmetic. If the system has integer coefficients, it is therefore possible to find a basis with vectors in $\mathbb{Q}^n$. You can now express the known positive real solution as a linear combination of the rational basis vectors. This linear combination is a continuous function of the coefficients of the basis vectors, therefore it is possible to change the coefficients to rational numbers without violating the positiveness of the solution vector. And if there is a positive rational solution, there is also a positive integer solution - simply multiply the solution vector with the LCD of its elements.