  Let $E/\mathbf{Q}$ be a CM elliptic curve and let $p\geq 5$ be a prime of
good ordinary reduction for $E$. Suppose that $L(E,s)$ vanishes at $s=1$ and
has sign $+1$ in its functional equation, so in particular ${\rm
ord}_{s=1}L(E,s)\geq 2$. In this paper we slightly modify a construction of
Darmon--Rotger to define a generalised Kato class $\kappa_p\in{\rm
Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of
Kolyvagin's result: \[ \kappa_p\neq 0\quad\Longrightarrow\quad{\rm
dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm
dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $\kappa_p\neq 0$
\emph{if and only if} the restriction map \[ {\rm
Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat\otimes\mathbf{Q}_p \] is
nonzero. The proof of these results, which extend and strenghten similar
results of the author with Hsieh in the non-CM case, exploit a new link between
the nonvanishing of generalised Kato classes and a main conjecture in
anticyclotomic Iwasawa theory.
