An immersion \(F:M^2\to \mathbb{R}^3\), oriented by a unit normal field \(n:M^2\to S^2\), induces a lift \(F= (f,n): M^2\to \mathbb{R}^3\times S^2=:\Lambda\) into the space of contact elements. \(\Lambda\) can be considered as the underlying space for Laguerre geometry: the geometry of the group is those transformations that map oriented planes in \(\mathbb{R}^3\) to oriented planes and oriented spheres (including the ``point spheres'' of radius 0) to oriented spheres. As such, the Laguerre group is a subgroup of the group of Lie sphere transformations which leave the space of spheres and planes invariant. As the principal curvature directions of a surface are invariant under Lie transformations, the notion of curvature line coordinates makes sense in Laguerre geometry. For a ``Legendre immersion'' \(F=(f,n)\), the role of the induced metric of an immersion is taken by the invariant bilinear form \(\frac{H^2-K}{K^2} dn\cdot dn\). Thus, in Laguerre geometry, a surface is called ``isothermal'' if it allows curvature line coordinates which are conformal with respect to the third fundamental form \(dn\cdot dn\). Considering Laguerre geometry as a subgeometry of Lie sphere geometry and using the method of moving frames, the authors give their main results: the only Laguerre-applicable surfaces are the Laguerre-isothermal surfaces -- in the reviewer's opinion, this might be extendable to an analog of the Darboux transformation for isothermic surfaces in Möbius geometry; the only second order Laguerre-deformable surfaces are the Laguerre-isothermal surfaces; the Pfaffian differential system for the Laguerre-isothermal surfaces depends on four functions of one variable -- again, similar to the case of isothermic surfaces in Möbius geometry.