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

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in $2{+}1$ dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form $v_t=v_xv_y-\partial^{-1}_x\,\partial_y[v_y+v^2_x]$, where the formal integral $\partial^{-1}_x$ becomes the asymmetric integral $-\int_x^{\infty}dx'$. We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function $f(X,Y)$ over a parabola in the plane $(X,Y)$ can be expressed in terms of the integrals of $f(X,Y)$ over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.