The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$

Question

There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ring $\mathbb{Z}$. Specifically, the issue arises from the first equation in the image. \begin{equation} \left(\begin{array}{cccc} a_{11} & a_{21} & \cdots & a_{m 1} \\ a_{12} & a_{22} & \cdots & a_{m 2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1 n} & a_{2 n} & \cdots & a_{m n} \end{array}\right)\left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_m \end{array}\right)=\left(\begin{array}{c} 0(modz_1) \\ 0(modz_2) \\ \vdots \\ 0(modz_n) \end{array}\right) \end{equation} Considering all numbers are in $\mathbb{Z}$, I know that the first equation given in the image must have a non-zero solution in $\mathbb{Z}$, such as taking the product of $z_1z_2\cdots z_n$. Essentially, this corresponds to whether the second equation has a solution, where $k_i$ can be any integer. \begin{equation} \left(\begin{array}{cccc} a_{11} & a_{21} & \cdots & a_{m 1} \\ a_{12} & a_{22} & \cdots & a_{m 2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1 n} & a_{2 n} & \cdots & a_{m n} \end{array}\right)\left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_m \end{array}\right)=\left(\begin{array}{c} k_1z_1 \\ k_2z_2 \\ \vdots \\ k_nz_n \end{array}\right) \end{equation} However, on this basis, I also want to know if there are solutions with an odd sum (or, in other words, solutions with a 1-norm as odd). So, there is the third equation, where the coefficient matrix is augmented with a row of 1's, and the non-homogeneous term corresponds to an odd number. This corresponds to whether the first equation has a solution with an odd sum. I wonder if there are any friends who are familiar with this and could provide some guidance. \begin{equation} \left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ a_{11} & a_{21} & \cdots & a_{m 1} \\ a_{12} & a_{22} & \cdots & a_{m 2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1 n} & a_{2 n} & \cdots & a_{m n} \end{array}\right)\left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_m \end{array}\right)=\left(\begin{array}{c} 2 t_0+1 \\ k_1 z_1 \\ k_2 z_2 \\ \vdots \\ k_n z_n \end{array}\right) \end{equation}