Context. Evolved stars with planets are crucial to understanding the dependency of the planet formation mechanism on the mass and metallicity of the parent star and to studying star-planet interactions. Aims: We present two evolved stars (HD 103485 and BD+03 2562) from the Tracking Advanced PlAnetary Systems (TAPAS) with HARPS-N project devoted to RV precision measurements of identified candidates within the PennState - Toruń Centre for Astronomy Planet Search. Methods: The paper is based on precise radial velocity (RV) measurements. For HD 103485 we collected 57 epochs over 3317 days with the Hobby-Eberly Telescope (HET) and its high-resolution spectrograph and 18 ultra-precise HARPS-N data over 919 days. For BD+03 2562 we collected 46 epochs of HET data over 3380 days and 19 epochs of HARPS-N data over 919 days. Results: We present the analysis of the data and the search for correlations between the RV signal and stellar activity, stellar rotation, and photometric variability. Based on the available data, we interpret the RV variations measured in both stars as Keplerian motion. Both stars have masses close to Solar (1.11 M⊙ HD 103485 and 1.14 M⊙ BD+03 2562), very low metallicities ([Fe/H] = - 0.50 and - 0.71 for HD 103485 and BD+03 2562), and both have Jupiter planetary mass companions (m2sini = 7 and 6.4 MJ for HD 103485 and BD+03 2562 resp.) in close to terrestrial orbits (1.4 au HD 103485 and 1.3 au BD+03 2562) with moderate eccentricities (e = 0.34 and 0.2 for HD 103485 and BD+03 2562). However, we cannot totally rule-out the possibility that the signal in the case of HD 103485 is due to rotational modulation of active regions. Conclusions: Based on the current data, we conclude that BD+03 2562 has a bona fide planetary companion while for HD 103485 we cannot totally exclude the possibility that the best explanation for the RV signal modulations is not the existence of a planet but stellar activity. If the interpretation remains that both stars have planetary companions, they represent systems orbiting very evolved stars with very low metallicities, a challenge to the conditions required for the formation of massive giant gas planets. Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.