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A $(d-\\lambda)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called Kneser transversal. If in addition $L$ contains $(d-\\lambda)+1$ points of $X$, then $L$ is  called complete Kneser transversal.In this paper, we present various results on the existence of (complete) Kneser transversals for $\\lambda =2,3$.  In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. 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