Free algebras of a unary variety with Mal'tsev's operation that satisfies the Pixley conditions

In this paper we consider the variety $VP$ of algebras with one unary and one ternary operation $p$ that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety $VP$ and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety $VP$ defined by the identities $p(p(x,y,z),y,z)=p(x,y,z)$ and $p(x,y,p(x,y,z))=p(x,y,z)$.