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Applying Lie's theory, we show that any $\mathcal{C}^\omega$ hypersurface$M^5 \subset \mathbb{C}^3$ in the class $\mathfrak{C}_{2,1}$ carriesCartan-Moser chains of orders $1$ and $2$.  Integrating and straightening any order $2$ chain at any point $p \in M$ tobe the $v$-axis in coordinates $(z, \zeta, w = u + i\, v)$ centered at $p$, weshow that there exists a (unique up to 5 parameters) convergent change ofcomplex coordinates fixing the origin in which $\gamma$ is the $v$-axis so that$M = \{u=F(z,\zeta,\overline{z},\overline{\zeta},v)\}$ has Poincar\'e-Moserreduced equation: \begin{align} u & = z\overline{z} +\tfrac{1}{2}\,\overline{z}^2\zeta + \tfrac{1}{2}\,z^2\overline{\zeta} +z\overline{z}\zeta\overline{\zeta} +\tfrac{1}{2}\,\overline{z}^2\zeta\zeta\overline{\zeta} +\tfrac{1}{2}\,z^2\overline{\zeta}\zeta\overline{\zeta} +z\overline{z}\zeta\overline{\zeta}\zeta\overline{\zeta} \\ & + 2{\rm Re} \{z^3\overline{\zeta}^2 F_{3,0,0,2}(v) + \zeta\overline{\zeta} (3\,{z}^2\overline{z}\overline{\zeta} F_{3,0,0,2}(v) ) \} \\ & + 2{\rm Re} \{z^5\overline{\zeta} F_{5,0,0,1}(v) + z^4\overline{\zeta}^2 F_{4,0,0,2}(v) +z^3\overline{z}^2\overline{\zeta} F_{3,0,2,1}(v) +z^3\overline{z}\overline{\zeta}^2 F_{3,0,1,2}(v) + z^3{\overline{\zeta}}^3F_{3,0,0,3}(v) \} \\ & + z^3\overline{z}^3 {\rm O}_{z,\overline{z}}(1) + 2{\rmRe} ( \overline{z}^3\zeta {\rm O}_{z,\zeta,\overline{z}}(3) ) +\zeta\overline{\zeta}\, {\rm O}_{z,\zeta,\overline{z},\overline{\zeta}}(5).\end{align}  The values at the origin of Pocchiola's two primary invariants are: \[ W_0 =4\overline{F_{3,0,0,2}(0)}, \quad\quad J_0 = 20\, F_{5,0,0,1}(0). \]  The proofs are detailed, accessible to non-experts. The computer-generatedaspects (upcoming) have been reduced to a minimum.

On Convergent Poincar\'e-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds