norm[v:{__}]:=(Sqrt[Plus@@(((#)^2)&/@v)])

timeNow=AbsoluteTime[];
earthPosition=AstronomicalData["Earth" ,"Position"]
earthVelocity=(AstronomicalData["Earth" ,{"Position",DateList[timeNow+3600]}]-
AstronomicalData["Earth" ,{"Position",DateList[timeNow-3600]}])/(2*3600)
earthMass=AstronomicalData["Earth" ,"Mass"]
sunMass=AstronomicalData["Sun" ,"Mass"]

<<PhysicalConstants`

m={earthMass,sunMass}
r0={earthPosition[[{1,2}]],{0,0}}
v0={earthVelocity[[{1,2}]],{0,0}}
zeroPointIndex=-1;
κ=GravitationalConstant[[1]]
coords={x,y}

pocetTeles=Length[m];
silyNa[ii_][t_]:=Table[κ*m[[ii]]*m[[jj]]*(-r[ii][t]+r[jj][t])/((norm[(r[ii][t]-r[jj][t])])^3),{jj,1,pocetTeles}]
rovVect=Drop[Table[r[ii]''[t]==Total[silyNa[ii][t]]/m[[ii]],{ii,1,pocetTeles}],{zeroPointIndex}]/.r[zeroPointIndex]->(0&)

rov2=rovVect/.r[ii_Integer]:>Through[coords[ii]]/.Through[coords[ii_]][t]:>Through[Through[coords[ii]][t]]/.Derivative[n_][f:{__}][t]:>(Derivative[n][#][t]&/@f)

rov3=rov2/.Derivative[r_][v:{__}][t_]:>(Derivative[r][#][t]&/@v)

rov4=rov3/.(left:{__})==(right:{__})/;Length[left]==Length[right]:>((Equal@@#&)/@({left,right}\[Transpose]))

rov4=Thread/@rov3

okrPodm1=Table[coords[[iCoord]][iR][0]==r0[[iR,iCoord]],{iR,1,Length[r0]},{iCoord,1,Length[coords]}]

okrPodm1=Drop[Thread[Through[Through[coords[#]][0]]==r0[[#]]]&/@Range[Length[r0]],{zeroPointIndex}]

okrPodmDer1=Drop[Thread[Through[Thread[Derivative[1][Through[coords[#]]]][0]]==v0[[#]]]&/@Range[Length[r0]],{zeroPointIndex}]

proms=Union[Cases[rov4,_Symbol[_Integer][t],-1]]/.a_[t]:>a

Flatten[{rov4,okrPodm1,okrPodmDer1}]

tmax=370137600;
resNahr=NDSolve[Flatten[{rov4,okrPodm1,okrPodmDer1}],proms,{t,0,tmax}][[1]]

ParametricPlot[(Through@(Through@coords[1])[t])/.resNahr,{t,0,tmax}]