I have an elliptic curve $$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I have a system (Sage or MAGMA) which can find all integer solutions of an elliptic curve $$ y'^2 + a_1x'y' + a_3 = x'^3 + a_2x'^2 + a_4x' + a_6 $$ which corresponds to the special case $c_1=c_2=1.$ Since I'm working over $\mathbb{Q}$ there is no trouble with the characteristic, and I can find a change of variables $x'=\alpha x+\beta,\ y'=\gamma y+\delta$ to obtain this form. But I'm looking for integral solutions to $(x,y)$ rather than $(x',y')$. Can I still solve this problem with the (black-box) solver?
Alternatively, are there solvers that take $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and find all integer solutions?
Example
Suppose I'm trying to find integer solutions of $3y^2 - 1 = x^3 + x.$ Let $x'=x/3$ and $y'=y/3,$ then the new curve is $y'^2 - 1/27 = x'^3 + x'/9$. But the integer solution $(x,y)=(1,1)$ in the original corresponds to the rational solution $(x',y')=(1/3,1/3)$ in the second, and hence would be missed if looking only for the integer solutions to the second equation.
Suppose $(X,Y)$ is an integral point on the elliptic curve $$c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6.$$ Then plugging in $X$ and $Y$ and multiplying through by $c_1^3c_2^2$ shows that $$(c_1^2c_2Y)^2+a_1(c_1c_2X)(c_1^2c_2Y)+a_3=(c_1c_2X)^3+a_2c_1(c_1c_2X)^2+a_4c_1^2c_2(c_1c_2X)+a_6c_1^3c_2^2,$$ so the point $(c_1c_2X,c_1^2c_2Y)$ is an integral point on the elliptic curve $$y^2+a_1xy+a_3=x^3+a_2c_1x^2+a_4c_1^2c_2x+a_6c_1^3c_2^2.$$ Now use your black box to find integral points on this elliptic curve, and then for each integral point check whether the $x$- and $y$-coordinates are divisible by $c_1c_2$ and $c_1^2c_2$, respectively, to determine whether it corresponds to an integral point on the original elliptic curve.