  This paper introduces a complex representation for spacelike surfaces in the
Lorentz-Minkowski space $L^4$, based in two complex valued functions which can
be assumed to be holomorphic or anti-holomorphic. When the immersion is
contained in quadrics of $L^4$, the representation then allows us to obtain
interesting partial differential equations with holomorphic or anti-holomorphic
parameters, within which we find the partial Riccati Equation. Using then
theory of holomorphic complex functions we construct explicitly new local
solutions for those PDEs together with its associated geometric solutions. So,
several explicit examples are given. As geometric consequence, through of our
approach we characterize all conformal totally umbilical spacelike immersions
into $L^4$, and in addition, we also show that for each conformal immersion in
$L^4$ which satisfies the partial Riccati equation there exists a Bryant
immersion in $H^3$, both immersions being congruent by a translation vector.
