Exact solutions to two contact problems with unknown contact domains on the elliptic punch penetration in the transversely isotropic elastic half-space are compared. In the first one - a “traditional problem” - the half-space boundary is parallel to the isotropy planes. There can be an axial symmetry case for a circular punch. In the second - a “nontraditional problem” - the half-space boundary is perpendicular to the isotropy planes. Here, the axial symmetry is impossible in principle: even for a circular punch, the contact domain will be elliptical. The contact domains and the forces required for the penetration of the punch at a given value are compared for both problems. The transversely isotropic body model is applicable for describing lots of materials which elastic parameters have been measured in recent decades: composites, ceramics, carbon fiber, graphite-epoxy, glass-epoxy, wood, aged concretes, some metals (titanium, cobalt, zinc), some semiconductors and rocks. They are widely used in the engineering and industry.