  The spin-1 Heisenberg antiferromagnet on a triangular lattice with the
nearest- and next-nearest-neighbor couplings, $J_1=(1-p)J$ and $J_2=pJ$, $J>0$,
is studied in the entire range of the parameter $p$. Mori's projection operator
technique is used as a method which retains the rotation symmetry of spin
components and does not anticipate any magnetic ordering. For zero temperature
four second-order phase transitions are observed. At $p\approx 0.038$ the
ground state is transformed from the long-range ordered $120^\circ$ spin
structure into a state with short-range ordering, which in its turn is changed
to a long-range ordered state with the ordering vector ${\bf
Q^\prime}=\left(0,-\frac{2\pi}{\sqrt{3}}\right)$ at $p\approx 0.2$. For
$p\approx 0.5$ a new transition to a state with a short-range order occurs.
This state has a large correlation length which continuously grows with $p$
until the establishment of a long-range order happens at $p \approx 0.65$. In
the range $0.5<p<0.96$, the ordering vector is incommensurate. With growing $p$
it moves along the line ${\bf Q'-Q}_1$ to the point ${\bf
Q}_1=\left(0,-\frac{4\pi}{3\sqrt{3}}\right)$ which is reached at $p\approx
0.96$. The obtained state with a long-range order can be conceived as three
interpenetrating sublattices with the $120^\circ$ spin structure on each of
them.
