Let f(x) be a smooth strictly convex solution of \( \det \left( \tfrac{\partial ^{2}f}{\partial x_{i}\partial x_{j}}\right) =\exp \left\{ \sum _{i=1}^n- a_i\tfrac{\partial f}{\partial x_{i}} +\sum _{i=1}^n b_ix_i+c \right\} \) defined on \(\mathbb {R}^{n}\) , where \(a_i\) , \(b_i\) and c are constants. Then, the graph \(M_{\nabla f}\) of \(\nabla f\) is a space-like translating soliton for the mean curvature flow in pseudo-Euclidean space \({\mathbb R}^{2n}_{n}\) with the indefinite metric \(\sum dx_idy_i\) . In this paper, we further investigate the properties of its entire solutions for some special translating vectors \(T=(a_1,\cdots ,a_n;b_1,\cdots ,b_n)\) . In particular, we demonstrate a non-existence result of entire Lagrangian translating soliton on \(\mathbb {R}^{2}\) if T is a lightlike vector with \((a_1,a_2)\ne 0,\;(b_1,b_2)\ne 0\) , and we find that the equation with \(a_1=a_2=0\) has a close relation to the Laplacian equation on Euclidean space \(\mathbb {R}^3\) .
