$\begin{array}{l} {x^4} - {y^4} + {z^4} + 2{x^2}{z^2} + 3{x^2} + 4{z^2} + 1 = 0\\ \Leftrightarrow \left( {{x^4} + 2{x^2}{z^2} + {z^4}} \right) + 3{x^2} + 4{z^2} + 1 = {y^4}\\ \Leftrightarrow {\left( {{x^2} + x} \right)^2} + 3{x^2} + 4x + 1 = {y^4}\\ \Rightarrow {y^4} > {\left( {{x^2} + x} \right)^2}\\ + {x^4} - {y^4} + {z^4} + 2{x^2}{z^2} + 3{x^2} + 4{z^2} + 1 = 0\\ \Leftrightarrow {y^4} = {x^4} + 2{x^2}{z^2} + {z^4} + 3{x^2} + 4{z^2} + 1\\ \Rightarrow {x^4} + 2{x^2}{z^2} + {z^4} + 3{x^2} + 4{z^2} + 1 < {x^4} + {x^2}{z^2} + {z^4} + 4{z^2} + 4{x^2} + 4\\ = {\left( {{x^2} + {z^2} + 2} \right)^2}\\ \Rightarrow {\left( {{x^2} + {z^2}} \right)^2} < {y^4} < {\left( {{x^2} + {z^2} + 2} \right)^2}\\ \Rightarrow {y^4} = {\left( {{x^2} + {z^2} + 1} \right)^2}\\ \Rightarrow {x^4} + 2{x^2}{z^2} + {z^4} + 3{x^2} + 4{z^2} + 1 = {x^4} + {z^4} + 1 + 2{z^2}{x^2} + 2{x^2} + 2{z^2} + 1\\ \Leftrightarrow {x^2} + 2{z^2} = 0\\ \Rightarrow \left\{ \begin{array}{l} x = 0\\ z = 0 \end{array} \right. \Rightarrow {y^4} = 1 \Rightarrow y = \pm 1\\ \Rightarrow \left( {x;y;z} \right) = \left( {0;1;0} \right),\left( {0; - 1;0} \right) \end{array}$
