The paper describes the design of a microscope for studying a betatron radiation source based on the PEARL femtosecond laser complex in the SXR and EUV wavelength range. The main optical element of the microscope is a spherical Schwarzschild objective a x5 magnification. The device allows to study the size and spatial structure of the interaction area of laser radiation with matter, at a selected wavelength in the EUV or SXR range with a resolution of δx = 2.75 microns. The operation wavelength (=13.5 nm) is set by multilayer X-ray mirrors. Thin-film absorption filters are used to suppress the background component of the signal.