We consider an integrodifferential equationwhose leading part coincides with the wave operator,while the lower part includes the nonlinear term $ q({\mathbf{x}})u^{m} $ with $ m>1 $ and a nonlinear integral operator.This operator models the memory of a mediumand includes a variable coefficient $ p({\mathbf{x}}) $ .For the original equationwe study the structure of the solution tothe Cauchy problem with zero initial dataand a pulsating source localized at some point $ ({\mathbf{y}},0) $ of the four-dimensional space of variables $ ({\mathbf{x}},t) $ .We assume that $ q({\mathbf{x}}) $ and $ p({\mathbf{x}}) $ are compactly supported functionswith supports lying in the ball $ B_{0} $ centered at the originand bounded by some sphere $ S_{0} $ ,while the point $ {\mathbf{y}} $ lies on a sphere $ S $ concentric with $ S_{0} $ but of larger radius.The point $ {\mathbf{y}} $ is a parameter of the problemand can run over the whole sphere $ S $ .We study the inverse problem of determining the functions $ q({\mathbf{x}}) $ and $ p({\mathbf{x}}) $ on $ B_{0} $ .For that we use the following information.For every point $ {\mathbf{y}} $ on the sphere $ S $ and a point $ {\mathbf{x}} $ on a certain part of the same spherewe define the solution to the Cauchy problemfor the original integrodifferential equationfor the moments of time close tothe arrival of the wave from the source at $ {\mathbf{y}} $ to $ {\mathbf{x}} $ .We show thatthis inverse problem reduces totwo identical integral geometry problemson the family of straight lineswith a prescribed weight functioninvariant under all rotations about the center of the ball $ B_{0} $ .We establish a uniqueness theoremand propose a method for solving these problems.
