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In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bfE}%_{1}^{3}$ that satisfy a linear Weingarten condition of type$\kappa_{1}=m\kappa_{2}+n$, where $m$ and $n$ are constant and $\kappa_{1}$ and$\kappa_{2}$ denote the principal curvatures at each point of the surface. Westudy the family of surfaces foliated by a uniparametric family of circles inparallel planes. We prove that the surface must be rotational or the surface ispart of the family of Riemann examples of maximal surfaces ($m=-1$, $n=0$).Finally, we consider the class of rotational surfaces for the case $n=0$,obtaining a first integration if the axis is timelike and spacelike and acomplete description if the axis is lightlike.

Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures