  Idempotent left nondegenerate solutions of the Yang-Baxter equation are in
one-to-one correspondence with twisted Ward left quasigroups, which are left
quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using
combinatorial properties of the Cayley kernel and the squaring mapping, we
prove that a twisted Ward left quasigroup of prime order is either
permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups
$(X,*)$ are obtained by twisting the left division operation in groups (that
is, they are of the form $x*y=\psi(x^{-1}y)$ for a group $(X,\cdot)$ and its
automorphism $\psi$), and they correspond to idempotent latin solutions. We
solve the isomorphism problem for idempotent latin solutions.
