2. Find all right-angled triangles with integer side lengths, in which the hypotenuse is one unit longer than one of the legs.
5.II.2. Let the legs of a right triangle have lengths $a$ and $b$, and the length of its hypotenuse be $a+1$. Then $a^{2}+b^{2}=(a+1)^{2}$, from which $b^{2}=2 a+1$. Therefore, $b=2 k+1$, so $4 k^{2}+4 k+1=2 a+1$, or $a=2 k(k+1)$.

Answer: all triangles with legs $2 k+1$ and $2 k(k+1)$ and hypotenuse $2 k^{2}+2 k+1$, where $k$ is any natural number.