This is probably not as explicit as what you requested, but there are recipes for how to build a Morse function on any smooth hypersurface in $CP^3$ with no index 1 or 3 critical points. There will be exactly 24 critical points, and any gradient vector field for this Morse function will be of the sort you asked about.
Such Morse functions are due to Rudolph (Topology 14,301-303. 1975), Mandlebaum-Moishezon (Topology 14, 23-40. 1976) and Harer (Math Annalen 238, 51-158, 1978). The first two use a Morse function coming from the distance (squared) of the hypersurface to a point in the complement, in the fashion of the Morse theory proof of the Lefschetz hyperplane theorem (cf. Andreotti-Frankel, Annals of Math 69 713–717, 1959, Bott, Michigan Math Journal 6 (3): 211–216, 1959, or Milnor's book on Morse theory). The latter uses a Lefschetz pencil. In principle, all of these could be converted to explicit formulas, but I don't know how, or if anyone has written this down.
You can find pictures of the resulting handlebody structure, with only index 0,2 and 4 handles (so exactly 24 of them) in Harer-Kas-Kirby (Mem. AMS 1986). You can find similar pictures for all elliptic surfaces in the book of Gompf-Stipsicz.