  Let $p\ge 7$ be a prime number and $f$ a normalized eigen-newform with good
reduction at $p$ such that its $p$-th Fourier coefficient vanishes. We
construct a rank-two Euler system attached to the $p$-adic realization of the
symmetric square motive of $f$. Furthermore, we show that the non-triviality is
guaranteed by the non-vanishing of the leading term of the relevant $L$-value
and the non-vanishing of a certain $p$-adic period modulo $p$.
