Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So this surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A result of Peter Shalen guarantees that a compact, connected, orientable, irreducible 3-manifold $M$ [such a manifold is known as Haken manifold] will have a non-trivial separating incompressible surface provided that $H_1(M;\mathbb Q)$ is carried by the boundary of M and that some boundary component of M has genus $> 1$.

Qestion 1: Does there exist any closed Haken 3-manifold with all its incompreesible surfaces are non-separating.

My intuition is NO. If we consider $S^1\times S^1\times S^1$, then all its incompressible surfaces has to be torus since this is the only surface with abeleian fundamental group. And then any incompressible torus transverse to the fiber (thinking it as a torus bundle over circle) has to intersect it with a non-trivial simple closed curve. And by some more argument we can prove that it is non separating since it can't separate any fiber.

Qestion 2: What if the fundamental group is non-abelian? Moreover what if we consider Hyperbolic Haken manifold, then can anyone give me an explicit example or reference of such a manifold whose all incompressible surfaces are non-separating?

Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A result of Peter Shalen guarantees that a compact, connected, orientable, irreducible 3-manifold $M$ [such a manifold is known as a Haken manifold] will have a non-trivial separating incompressible surface provided that $H_1(M;\mathbb Q)$ is carried by the boundary of M and that some boundary component of M has genus $> 1$.

Question 1: Does there exist any closed Haken 3-manifold with all its incompressible surfaces are non-separating.

My intuition is NO. If we consider $S^1\times S^1\times S^1$, then all its incompressible surfaces has to be torus since this is the only surface with abelian fundamental group. And then any incompressible torus transverse to the fiber (thinking it as a torus bundle over circle) has to intersect it with a non-trivial simple closed curve. And by some more argument we can prove that it is non separating since it can't separate any fiber.

Question 2: What if the fundamental group is non-abelian? Moreover what if we consider a hyperbolic Haken manifold, then can anyone give me an explicit example or reference of such a manifold whose all incompressible surfaces are non-separating?