New primitives $3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$

$$3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$$ A table of primitives known to me $(a, b, c)$, $a \in{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}^+$ $$ \boxed{\begin{array} {|r|r|r|r|}\hline №(a,b,c)& a_n & b_n & c_n \\ \hline S_1 & 0 & 0 & 0 \\ \hline S_2 & 0 & -3 & 3 \\ \hline S_n & -380010404710181340279 & 189768844608364006623 & 190083988306799187048 \\ \hline S_4 & ? & ? & ? \end{array}} $$ During my inspection, I found that there are no more integers up to heights. [$a<1,71×10^7$, $b<3,63×10^8$].

To find this solution, I randomly generated a rational point from a lost parametric solution for an elliptic curve. $$a=\frac{141439558667661294516150296481754021287}{46994659393379475275019815927126707841}$$ $$b=-\frac{52129288259885406377825559923882427}{46994659393379475275019815927126707841}$$ $$c=-\frac{141270322102587708845544476741296591380}{46994659393379475275019815927126707841}$$

After checking other related points, I managed to find the same integer point. $\boxed{S(a,b,c)=-380010404710181340279, 189768844608364006623, 190083988306799187048}$ It's not completely clear if this solution is the smallest of the range, or if it's a brilliant coincidence.

are there other nontrivial solutions (primitives a,b,c) for this diophantine equation, and is it possible to obtain less?

English is not my native language, I would appreciate your help in correcting errors.