  Motivated by the classical Euler elastic curves, David A. Singer posed in
1999 the problem of determining a plane curve whose curvature is given in terms
of its position. We propound the same question in Lorentz-Minkowski plane,
focusing on spacelike and timelike curves. In this article, we study those
curves in $\mathbb{L}^2$ whose curvature depends on the Lorentzian
pseudodistance from the origin, and those ones whose curvature depends on the
Lorentzian pseudodistance through the horizontal or vertical geodesic to a
fixed lightlike geodesic. Making use of the notions of geometric angular
momentum (with respect to the origin) and geometric linear momentum (with
respect to the fixed lightlike geodesic) respectively, we get two abstract
integrability results to determine such curves through quadratures. In this
way, we find out several new families of Lorentzian spiral, special elastic and
grim-reaper curves whose intrinsic equations are expressed in terms of
elementary functions. In addition, we provide uniqueness results for the
generatrix curve of the Enneper's surface of second kind and for Lorentzian
versions of some well known curves in the Euclidean setting, like the Bernoulli
lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate
conics. We are able to get arc-length parametrizations of them and they are
depicted graphically.
