Although billions of COVID-19 vaccines have been administered, too many people remain hesitant. Misinformation about the COVID-19 vaccines, propagating on social media, is believed to drive hesitancy towards vaccination. However, exposure to misinformation does not necessarily indicate misinformation adoption. In this paper we describe a novel framework for identifying the stance towards misinformation, relying on attitude consistency and its properties. The interactions between attitude consistency, adoption or rejection of misinformation and the content of microblogs are exploited in a novel neural architecture, where the stance towards misinformation is organized in a knowledge graph. This new neural framework is enabling the identification of stance towards misinformation about COVID-19 vaccines with state-of-the-art results. The experiments are performed on a new dataset of misinformation towards COVID-19 vaccines, called CoVaxLies, collected from recent Twitter discourse. Because CoVaxLies provides a taxonomy of the misinformation about COVID-19 vaccines, we are able to show which type of misinformation is mostly adopted and which is mostly rejected.