Let $M,N\subset \mathbb C^2$ be real-analytic strictly pseudoconvex real hypersurfaces, let $p\in M$, let $q\in N$, and equip $M$ and $N$ with their induced real-analytic CR structures. Let $G:=SU(2,1)$, let $P\subset G$ be the parabolic subgroup stabilizing a complex null line in the standard model, and let
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\begin{align*}
\pi_M:\mathcal G_M\to M
\end{align*}
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and
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\begin{align*}
\pi_N:\mathcal G_N\to N
\end{align*}
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be the canonical real-analytic Cartan CR principal $P$-bundles with Cartan connections
\begin{align*}
\kappa_M:\mathcal G_M\to \mathbb K
\end{align*}
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and
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\begin{align*}
\kappa_N:\mathcal G_N\to \mathbb K
\end{align*}
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be their Cartan curvature functions, where $\mathbb K$ is the fixed normal CR curvature representation for Cartan geometries of type $(SU(2,1),P)$.
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For a Cartan geometry $(\mathcal G,\omega)$ and a real-analytic map $F:\mathcal G\to W$ into a finite-dimensional representation space $W$, write $\nabla^\omega F$ for the invariant derivative
for all integers $m\ge 0$, with $(\nabla^\omega)^0F:=F$.
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Then the germs $(M,p)$ and $(N,q)$ are locally CR equivalent by a real-analytic CR diffeomorphism sending $p$ to $q$ if and only if there exist frames $u\in(\mathcal G_M)_p$ and $v\in(\mathcal G_N)_q$ such that, for every integer $m\ge 0$,
Equivalently, the full infinite Cartan curvature jets at $p$ and $q$ lie in the same orbit under the residual $P$-action on the sequence of model representation spaces.