  It is proved that for every surjective linear isometry $V$ on a perfect
Banach symmetric ideal $\mathcal C_E\neq \mathcal C_2$ of compact operators,
acting in a complex separable infnite-dimensional Hilbert space $\mathcal H$
there exist unitary operators $u$ and $v$ on $\mathcal H$ such that $V(x)=uxv$
or $V(x) = ux^tv$ for all $x\in \mathcal C_E$, where $x^t$ is the transpose of
an operator $x$ with respect to a fixed orthonormal basis in $\mathcal H$. In
addition, it is shown that any surjective 2-local isometry on a perfect Banach
symmetric ideal $\mathcal C_E \neq \mathcal C_2$ is a linear isometry on
$\mathcal C_E$.
