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As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form ofintegrable dispersionless multidimensional PDEs is non-local, and the properchoice of integration constants should be the one dictated by the associatedInverse Scattering Transform (IST). Using the recently made rigorous IST forvector fields associated with the so-called Pavlov equation$v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, we have recently esatablished that, inthe nonlocal part of its evolutionary form $v_{t}=v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$, the formalintegral $\partial^{-1}_{x}$ corresponding to the solutions of the Cauchyproblem constructed by such an IST is the asymmetric integral$-\int_x^{\infty}dx'$. In this paper we show that this results could be guessedin a simple way using a, to the best of our knowledge, novel integral geometrylemma. Such a lemma establishes that it is possible to express the integral ofa fairly general and smooth function $f(X,Y)$ over a parabola of the $(X,Y)$plane in terms of the integrals of $f(X,Y)$ over all straight lines nonintersecting the parabola. A similar result, in which the parabola is replacedby the circle, is already known in the literature and finds applications intomography. Indeed, in a two-dimensional linear tomographic problem with aconvex opaque obstacle, only the integrals along the straight linesnon-intersecting the obstacle are known, and in the class of potentials$f(X,Y)$ with polynomial decay we do not have unique solvability of the inverseproblem anymore. Therefore, for the problem with an obstacle, it is natural notto try to reconstruct the complete potential, but only some integralcharacteristics like the integral over the boundary of the obstacle. Due to theabove two lemmas, this can be done, at the moment, for opaque bodies having asboundary a parabola and a circle (an ellipse).

An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects