  For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a
2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq
2$ and $n$, a \textit{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$ is a
subset $S$ of $[n]\times [n]$ such that there are no elements ${\mathbf{a}}_1,
\ldots, {\mathbf{a}}_k$ in $S$ satisfying ${\mathbf{a}}_1+\cdots+{\mathbf{a}}_k
=\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and
positive integers $k\geq 2$ and $n$, we determine the maximum density of a
{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$. This is the first
investigation of the non-homogeneous sum-free set problem in higher dimensions.
