Eruptive stars are a class of young stellar objects that show an abrupt increase in luminosity. These burst-like episodes are thought to dominate the stellar accretion process during the class 0 to class I stage. We present an overview of a survey of seven episodically accreting protostars carried out to study their potentially complex circumstellar surroundings. The observations were performed with the instrument SPHERE, mounted at the Very Large Telescope. SPHERE is equipped with an extreme adaptive optics system that allows high-contrast imaging. We observed the eruptive stars in the $H$ band with the near-infrared imager IRDIS and used the polarimeter to extract the polarized light scattered from the stars' surroundings. We produced polarized light images for three FUor objects, Z CMa, V960 Mon, and FU Ori, and four EXor objects, XZ Tau, UZ Tau, NY Ori, and EX Lup. We calculated the intrinsic polarization fraction for all the observed stars. In all systems we registered scattered light from around the primary star. FU Ori and V960 Mon are surrounded by complex structures, including spiral-like features. In Z CMa, we detected a point source 0 to the northeast of the primary. Based on the astrometric measurements from archival Keck/NIRC2 data, we find this source to be a third member of the system. Furthermore, Z CMa displays an outflow that extends for thousands of au. Unlike the other EXor objects in our sample, XZ Tau shows bright, extended scattered light structures that are also associated with an outflow on a scale of hundreds of au. The other EXors show relatively faint disk-like structures in the immediate vicinity of the coronagraph. Each object shows a unique environment, but we classified the seven objects into three categories: systems with illuminated outflows, asymmetric arms, and faint disks. Asymmetric arms were only found around FUor objects, while faint disks seem to predominantly occur around EXors. Importantly, for Z CMa the detection of the faint extended structure calls into question previous interpretations of the system's dynamic state. The streamer that was associated with a fly-by object turned out to be part of a huge outflow extending 6000 au.
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