  The strong truncated Hamburger moment problem (STHMP) of degree
$(-2k_1,2k_2)$ asks to find necessary and sufficient conditions for the
existence of a positive Borel measure, supported on $\mathbb{R}\setminus
\{0\}$, such that $\beta_i=\int x^id\mu\; (-2k_1\leq i\leq 2k_2)$. Using the
solution of the truncated Hamburger moment problem and the properties of Hankel
matrices we solve the STHMP. Then, using the equivalence with the STHMP of
degree $(-2k,2k)$, we obtain the solution of the 2-dimensional truncated moment
problem (TMP) of degree $2k$ with variety $xy=1$, first solved by Curto and
Fialkow. Our addition to their result is the fact previously known only for
$k=2$, that the existence of a measure is equivalent to the existence of a flat
extension of the moment matrix. Further on, we solve the STHMP of degree
$(-2k_1,2k_2)$ with one missing moment in the sequence, i.e., $\beta_{-2k_1+1}$
or $\beta_{2k_2-1}$, which also gives the solution of the TMP with variety
$x^2y=1$ as a special case, first studied by Fialkow.
