Cases of Further Integrability of the Equation of Motion in the Presence of the Local Integral in the Space Model. I

A local integral that allows us to immediately determine the velocity field for certain specific initial conditions was introduced by Antonov and Shamshiev in 1992. We consider cases where the integrability of the trajectory can be advanced further. We then obtain an analytical relationship between \(x\) , \(y\) , \(z\) , \(t\) on a given trajectory. Possible cases are classified and one of them is analyzed.