Let $S$ be a linear subspace of $\Bbb R^n$ having dimension $k<n$ and assume $S$ is described by $n-k$ linear equations with integer coefficients. Look at now the intersection $\Lambda=S\cap \Bbb Z^n$ - such an intersection is a lattice in $S$. Theoretically speaking, I already know the existence of a basis $\{v_1,\dots,v_k\} \in S$ such that $\Lambda=\big\{\lambda_1v_1+\cdots+\lambda_kv_k\,|\, \lambda_i\in \Bbb Z \text{ for all } i\big\}$. Of course, any $v_i$ has integer coefficients.
I am wondering if the basis $\big\{v_1,\dots,v_k\big\}$ can be completed to a basis $\big\{v_1,\dots,v_k,v_{k+1},\dots,v_{n}\big\}$ of $\Bbb R^n$ by adding other $n-k$ vectors (with integer coefficients) such that there is a matrix $g\in \text{SL}(n,\Bbb Z)$ such that $g(e_i)=v_i$ for any $i=1,\dots,n$, where $\{e_1,\dots,e_n\}$ denotes the standard basis.
In other words, I am wondering if $\{v_1,\dots,v_k\}$ can be completed to a primitive set of vectors generating the standard lattice $\Bbb Z^n$. By setting $V$ the $k\times n$ matrix having the vectors $v_i$ as columns, if I am not mistaken, this condition is equivalent to say that the gcd of the $k^{th}$ order minors is one. In this case, the matrix $V$ can be completed to a square matrix with determinant one - hence a matrix in $\text{SL}(n,\Bbb Z)$.
When $n=2$, this is always true. Indeed, let $qx-py=0$ a one-dimensional space in $\Bbb R^2$ - $p,q$ are taken coprime. Clearly $(p,q)$ satisfies the equation above. Taking any solution $(x_o,y_o)$ of the associated diophantine equation $qx-py=1$ (the solution exists because $p,q$ are coprime) we have two vectors $(p,q), \, (x_o,y_o)$ and they form a basis for the standard lattice. In other words, the basis $(p,q)$ is completed to a basis of $\Bbb Z^2$.
When $n=3$, the problem seems more subtle and I don't have an answer. I began with this example. I considered the plane $S$ given by the equation $x+y-2z$ in $\Bbb R^3$. The vectors $(2,0,1),\, (0,2,1)$ form a basis for $S$ and they belong $\Bbb Z^3$, clearly. However, they don't form a basis for the lattice $\Lambda=S\cap \Bbb Z^3$. Indeed, the vector $(1,1,1)$ doesn't belong to $\Bbb Z$-span of them. By chance, I noticed that $\big\{(2,0,1),\,(1,1,1)\big\}$ is a basis for the lattice $\Lambda$. Not only, by adding the vector $(0,-1,0)$, for instance, I can complete the latter basis to a basis of $\Bbb R^3$ and there is $g\in\text{SL}(3,\Bbb Z)$ such that $g(e_1)=(2,0,1),\, g(e_2)=(1,1,1),\, g(e_3)=(0,-1,0)$. I have made some other examples and they work similarly.
For a generic $n\ge3$, I don't know if there is an algorithmic method to find a basis for $\Lambda=S\cap \Bbb Z^n$ and if such a basis can be always complete to a basis for $\Bbb Z^n$, like in the $n=2$ case.
