Finding an analytic solution of a cubic equation

It comes with which roots correspond to which branches of a cubic root in the exact expression. For instance, consider the simpler

$$y^3 = z$$

Then my three solutions are $y=\sqrt[3]{z}$, $y=(-1)^{2/3}\sqrt[3]{z}$, and $y=(-1)^{-2/3}\sqrt[3]{z}$. Mathematica defines that $\sqrt[3]{-1} = \frac{1}{2} + \frac{\sqrt{3}}{2}i$. So if I track the first solution, as a curve in 3D of $(\Re(y),\Im(y),z)$ as I vary $z$ in $[-1,1]$, I'll see a strange behavior at 0, where I switch from $y$ being purely real, to $y$ heading off in an imaginary direction. You could argue that the "right" root to be taking would be the $\sqrt[3]{-1} = -1$ branch, and indeed if we switched from the first solution to the second at $z=0$ it will be analytic and all well. Similarly, the second solution at $z>0$ corresponds to the third at $z<0$, and correspondingly for the third and first.

The important element is that, due to the choice of branch, the solutions "switch roles" at a certain point.

In your specific equation, the roots also switch, so what "was" the first solution is now a different one. The precise behavior is a lot more complicated because of a different kind of singular behavior in the equations there leading to a discontinuity (not just nondifferentiability).

Calling Reduce with Cubics just called Solve indirectly. But Root specifies root by an ordering on the actual numerical value, not closed form structure, so it continued to select the same one.

To see what the behavior of the different solutions looks like, you can try running the below:

sols = Solve[y == z*y^3 + z, y];
{s1, s2, s3} = %[[All, 1, 2]];
Plot[{Re[s1], Re[s2], Re[s3]}, {z, -3, 3}, PlotStyle -> {Red, Green, Blue}]
Plot[{Im[s1], Im[s2], Im[s3]}, {z, -3, 3}, PlotStyle -> {Red, Green, Blue}]

See that the red solution on the left changes color to become blue; the blue and green solutions on the left both hit a pole to become green and red, in some order (up to choice of $\sqrt{-1} = \pm i$). The solution you wanted is Red on the left and Blue on the right