*Answer:*

$7$

Let the symbol ${\sum}_{\mathrm{cyc}}Q(x,y,z)$ indicate a sum where the other two terms are obtained by repeating twice the cyclic swap $x\to y\to z\to x$ hence, ${\sum}_{\mathrm{cyc}}Q(x,y,z)=Q(x,y,z)+Q(y,z,x)+Q(z,x,y)$.

Multiplying the equation by $\mathit{xyz}\ne 0$ and rearranging yields

$$P(x,y,z)=x{(x-1)}^{2}(y-z)+y{(y-1)}^{2}(z-x)+z{(z-1)}^{2}(x-y)=\sum _{\mathrm{cyc}}x{(x-1)}^{2}(y-z)=0.$$ |

Since $P$ vanishes for $x=y$, $y=z$, or $z=x$, it must be divisible by $(x-y)(y-z)(z-x)={\sum}_{\mathrm{cyc}}{x}^{2}(z-y)$. Since $P(x,y,z)$ is a polynomial of degree $4$ and ${\sum}_{\mathrm{cyc}}{x}^{2}(z-y)$ is a polynomial of degree $3$, the reminding factor must be linear

$$P(x,y,z)=\left(\sum _{\mathrm{cyc}}{x}^{2}(z-y)\right)\cdot \left(\mathit{ax}+\mathit{by}+\mathit{cz}+d\right).$$ |

Furthermore $\mathit{xy}-\mathit{xz}+\mathit{yz}-\mathit{yx}+\mathit{zx}-\mathit{zy}={\sum}_{\mathrm{cyc}}x(y-z)=0$, hence

$$\begin{array}{llll}\hfill P(x,y,z)& =\sum _{\mathrm{cyc}}\left({x}^{3}(y-z)-2{x}^{2}(y-z)+x(y-z)\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\sum _{\mathrm{cyc}}\left({x}^{3}(y-z)-2{x}^{2}(y-z)\right)+0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\left(\sum _{\mathrm{cyc}}{x}^{2}(z-y)\right)\cdot \left(\mathit{ax}+\mathit{by}+\mathit{cz}+d\right).\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$From this we can see that $a$ must be $-1$ so that ${x}^{2}(z-y)\cdot \mathit{ax}={x}^{3}(y-z)$ and similarly $b=c=-1$. Further, from ${x}^{2}(z-y)\cdot d=-2{x}^{2}(y-z)$, we obtain that $d=2$. Therefore,

$$P(x,y,z)=(x-y)(y-z)(z-x)(2-x-y-z)=0.$$ |

As we only look for pairwise distinct triples $(x,y,z)$, it must follow that $x+y+z=2$. It is easy to see that any triple with these properties solves the original equation as well.

To find the minimal value of the expression $|64x+19y+4z|$, subtract $4(x+y+z)-8=0$ to get

$$|64x+19y+4z|=|15\cdot (4x+y)+8|.$$ |

We are searching for an integer $4x+y$ that minimises the expression. The minimum is clearly attained for $4x+y=-1$, for instance, at $(x,y,z)=(-2,7,-3)$. Therefore, the result is $7$.