Azimuthal distributions of $eta$ and $pi^{0}$ mesons emitted at midrapidity in collisions of 1.9 AGeV $^{58}$Ni+$^{58}$Ni and 2 AGeV $^{40}$Ca+$^{nat}$Ca are studied as a function of the number of projectile-like spectator nucleons. The observed anisotropy corresponds to a negative elliptic flow signal for $eta$ mesons, indicating a preferred emission perpendicular to the reaction plane. In contrast, only small azimuthal anisotropies are observed for $pi^{0}$ mesons. This may indicate that $eta$ mesons freeze out earlier from the fire ball than pions.