We study the properties of bright and vortex solitons in thermal media with nonuniform thermal conductivity and\udhomogeneous refractive index, whereby the local modulation of the thermal conductivity strongly affects the entire\udrefractive index distribution. While regions where the thermal conductivity is increased effectively expel light, selftrapping\udmay occur in the regions with reduced thermal conductivity, even if such regions are located close to the\udmaterial boundary. As a result, strongly asymmetric self-trapped beams may form inside a ring with reduced thermal\udconductivity and perform persistent rotary motion. Also, such rings are shown to support stable vortex solitons,\udwhich may feature strongly noncanonical shapes.
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