Summary: On a spin Kähler manifold \(M^{2m}\) a new first integral \(Q_ \psi\) of the Kählerian twistor equations is presented. If the scalar curvature has a critical point then \(Q_ \psi\) vanishes. In case \(M^{2m}\) is closed, this fact provides a simple geometrical obstruction for Kählerian twistor spinors and, consequently, some new vanishing theorems. Twistor spinors with \(Q_ \psi \neq 0\) are investigated. Some invariants of the corresponding space of twistor spinors are constructed if the other basic first integral \(C_ \psi\) does not vanish, too. Each twistor spinor \(\psi\) with \(C_ \psi \neq 0\) and \(Q_ \psi \neq 0\) determines a foliation of \(M^{2m}\) whose leaves are totally geodesic immersed Kähler manifolds. It is shown that Kählerian twistor spinors of type \((r,s)\) can be interpreted as the minima of a certain functional. Some properties of Kählerian twistor spinors of the exceptional type \(r = (m + 2)/2\) are proved.