  L\'{e}vy walks are a particular type of continuous-time random walks which
results in a super-diffusive spreading of an initially localized packet. The
original one-dimensional model has a simple schematization that is based on
starting a new unidirectional motion event either in the positive or in the
negative direction. We consider two-dimensional generalization of L\'{e}vy
walks in the form of the so-called XY-model. It describes a particle moving
with a constant velocity along one of the four basic directions and randomly
switching between them when starting a new motion event. We address the
ballistic regime and derive solutions for the asymptotic density profiles. The
solutions have a form of first-order integrals which can be evaluated
numerically. For specific values of parameters we derive an exact expression.
The analytic results are in perfect agreement with the results of finite-time
numerical samplings.
