The boundary integral method is extended to derive closed integro-differential equations applicable to computation of the shape and propagation speed of a steadily moving spot and to the analysis of dynamic instabilities in the sharp boundary limit. Expansion of the boundary integral near the locus of traveling instability in a standard reaction-diffusion model proves that the bifurcation is supercritical whenever the spot is stable to splitting, so that propagating spots can be stabilized without introducing additional long-range variables.Localized structures in non-equlibrium systems (dissipative solitons) have been studied both in experiments and computations in various applications, including chemical patterns in solutions [1] and on surfaces [2], gas discharges [3] and nonlinear optics [4]. The interest to dynamic solitary structures, in particular, in optical [4] and gas discharge systems [5] has been recently driven by their possible role in information transmission and processing.A variety of observed phenomena can be reproduced qualitatively with the help of simple reaction-diffusion models with separated scales [6][7][8][9][10]. Extended models of this type included nonlocal interactions due to gas transport [9,11], Marangoni flow [12] or optical feedback [4,13]. A great advantage of scale separation is a possibility to construct analytically strongly nonlinear structures in the sharp interface limit. An alternative approach based on Ginzburg-Landau models supplemented by quintic and/or fourth-order differential (Swift-Hohenberg) terms [14] have to rely on numerics in more than one dimension.Dynamical solitary structures are most interesting from the point of view of both theory and potential applications. Existence of traveling spots in sharp-interface models is indicated by translational instability of a stationary spot [11]. This instability is a manifestation of a general phenomenon of parity breaking (Ising-Bloch) bifurcation [15,16] which takes a single stable front into a pair of counter-propagating fronts forming the front and the back of a traveling pulse. Numerical simulations, however, failed to produce stable traveling spots in the basic activator-inhibitor model, and the tendency of moving spots to spread out laterally had to be suppressed either by global interaction in a finite region [11] or by adding an extra inhibitor with specially designed properties [17].The dynamical problem is difficult for theoretical study, since a moving spot loses its circular shape, and a free-boundary problem is formidable even for simplest kinetic models. Numerical simulation is also problematic, due to the need to use fine grid to catch sharp gradients of the activator; therefore actual computations were carried out for moderate scale ratios. A large amount of numerical data, such as the inhibitor field far from the spot contour, is superfluous. This could be overcome if it was possible to reduce the PDE solution to local dynamics of a sharp boundary. Unfortunately, a purely local equation of front motio...