  For the truncated multidimensional moment problem we introduce a notion of a
canonical solution. Namely, canonical solutions are those solutions which are
generated by commuting self-adjoint extensions inside the associated Hilbert
space. It is constructed a 1-1 correspondence between canonical solutions and
flat extensions of the given moments (both sets may be empty). In the case of
the two-dimensional moment problem (with triangular truncations) a search for
canonical solutions leads to an algebraic system of equations. A notion of the
index $i_s$ of nonself-adjointness for a set of prescribed moments is
introduced. The case $i_s=0$ corresponds to flatness. In the case $i_s=1$ we
get explicit necessary and sufficient conditions for the existence of canonical
solutions. These conditions are valid for arbitrary sizes of truncations. In
the case $i_s=2$ we get either explicit conditions for the existence of
canonical solutions or a single quadratic equation with several unknowns.
Numerical examples are provided.
