= Solution
Apply Jacobi's reconstruction to the polynomial complete integral in <4/b/solution>. Its parameter derivatives are constant:
$$
\beta_x=S_{\alpha_x}=x-\frac{\alpha_x}{m}t,\qquad
\beta_y=y-\frac{\alpha_y}{m}t,\qquad
\beta_z=z-\frac{\alpha_z}{m}t+\frac12gt^2.
$$
Therefore
$$
\boxed{x(t)=\beta_x+\frac{\alpha_x}{m}t,\quad
y(t)=\beta_y+\frac{\alpha_y}{m}t,\quad
z(t)=\beta_z+\frac{\alpha_z}{m}t-\frac12gt^2.}
$$
At $t=0$ the constants $\boldsymbol\beta$ are the initial positions and $\boldsymbol\alpha$ are the initial <momenta>. Direct differentiation of $S$ gives $p_x=\alpha_x$, $p_y=\alpha_y$, $p_z=\alpha_z-mgt$, confirming the same motion. The six constants provide arbitrary initial data; the solution covers both ascent and descent, not merely one local square-root branch of the separated <action>.
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