In "Myths and Counterexamples of Mathematical Programming" myth "IP Myth 21" says:
The problem of finding $x\in \mathbb{Z}$ such that $Ax=b$, where $A\in\mathbb{Z}^{m\times n}$ and $b\in \mathbb{Z}^m$, is NP-complete.
Which is incorrect (as all myths in this excellent resource are).
The key to this myth is that there are no bounds on $x$. (If we add $x≥0$, we have the standard form of a linear integer program, which is NP-complete.) This is the problem of solving linear diophantine equations for which Brown provides simple, polynomial-time algorithms
Is there a name for this problem class in OR/Optimization?
Is there a name for this problem class some linear equality constraints over $\mathbb{R}$ are added?
Are you aware of any MILP solvers which find such subproblems and solve them using specialized routines?
Are problems of the same form but with $\min ||x||$, $\min c^Tx$, $\min \sum_i (c_i x_i)^2 $, $\min x^T D x$ or $\min x^T D x + c^Tx$ where D positive definite as objective still solvable in polynomial time?