Invariant measures for the geodesic flow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and well-studied subject. This paper continues an investigation into a $2$-dimensional analog of this flow for a $3$-manifold $N$. Namely, the article discusses $2$-dimensional surfaces immersed into $N$ whose product of principal curvature equals a constant $k$ between 0 and 1, surfaces which are called $k$-surfaces. The “$2$-dimensional” analog of the unit tangent bundle with the geodesic flow is a “space of pointed $k$-surfaces”, which can be considered as the space of germs of complete $k$-surfaces passing through points of $N$. Analogous to the $1$-dimensional lamination given by the geodesic flow, this space has a $2$-dimensional lamination. An earlier work  was concerned with some topological properties of chaotic type of this lamination, while this present paper concentrates on ergodic properties of this object. The main result is the construction of infinitely many mutually singular transversal measures, ergodic and of full support. The novel feature compared with the geodesic flow is that most of the leaves have exponential growth.