Applying Lies theory, we show that any $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in the class $mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at any point $p in M$ to be the $v$-axis in coordinates $(z, zeta, w = u + i, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $gamma$ is the $v$-axis so that $M = {u=F(z,zeta,overline{z},overline{zeta},v)}$ has Poincare-Moser reduced equation: begin{align} u & = zoverline{z} + tfrac{1}{2},overline{z}^2zeta + tfrac{1}{2},z^2overline{zeta} + zoverline{z}zetaoverline{zeta} + tfrac{1}{2},overline{z}^2zetazetaoverline{zeta} + tfrac{1}{2},z^2overline{zeta}zetaoverline{zeta} + zoverline{z}zetaoverline{zeta}zetaoverline{zeta} & + 2{rm Re} { z^3overline{zeta}^2 F_{3,0,0,2}(v) + zetaoverline{zeta} ( 3,{z}^2overline{z}overline{zeta} F_{3,0,0,2}(v) ) } & + 2{rm Re} { z^5overline{zeta} F_{5,0,0,1}(v) + z^4overline{zeta}^2 F_{4,0,0,2}(v) + z^3overline{z}^2overline{zeta} F_{3,0,2,1}(v) + z^3overline{z}overline{zeta}^2 F_{3,0,1,2}(v) + z^3{overline{zeta}}^3 F_{3,0,0,3}(v) } & + z^3overline{z}^3 {rm O}_{z,overline{z}}(1) + 2{rm Re} ( overline{z}^3zeta {rm O}_{z,zeta,overline{z}}(3) ) + zetaoverline{zeta}, {rm O}_{z,zeta,overline{z},overline{zeta}}(5). end{align} The values at the origin of Pocchiolas two primary invariants are: [ W_0 = 4overline{F_{3,0,0,2}(0)}, quadquad J_0 = 20, F_{5,0,0,1}(0). ] The proofs are detailed, accessible to non-experts. The computer-generated aspects (upcoming) have been reduced to a minimum.
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