Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that $$ 2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ? $$
A short motivation: there are many known counterexamples to the Hasse principle for cubic surfaces, for example, $15x^3 + 10y^3 + 4z^3 + 3t^3 = 0$. In this example, the sum $S$ of absolute values of the coefficients is $32$. This sum is even higher in all other counterexamples that I know. One may ask what is the counterexample for cubic surfaces with the smallest $S$. I have checked that Hasse principle holds for all cubic surfaces with $S\leq 11$. For the given equation, $S=12$, Magma returns that it is locally solvable (I checked for primes up to $100$). Direct search returns no non-trivial solutions with $\max(|x|,|y|,|z|)\leq 100$. Hence the question.
